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(a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) In the past, a federal law mandated that highway speed limits would be 55 mi/h. Use the conversion factor of part (a) to find this speed in kilometers per hour. (c) The maximum highway speed is now 65 mi/h in some places. In kilometers per hour, how much increase is this over the 55 mi/h limit?
The mass of the Sun is 1.99 x 1030 kg, and the mass of an atom of hydrogen, of which the Sun is mostly composed, is 1.67 X 10-27 kg. How many atoms are in the Sun?
One gallon of paint (volume =3.78 X 10-3 m3) covers an area of 25.0 m2. What is the thickness of the paint on the wall?
A pyramid has a height of 481 ft and its base covers an area of 13.0 acres (Fig. P1.32). If the volume of a pyramid is given by the expression V = ⅓ Bh, where B is the area of the base and h is the height, find the volume of this pyramid in cubic meters. (1 acre = 43 560 ft2)
The pyramid described in Problem 32 contains approximately 2 million stone blocks that average 2.50 tons each. Find the weight of this pyramid in pounds.
Assuming that 70% of the Earth’s surface is covered with water at an average depth of 2.3 mi, estimate the mass of the water on the Earth in kilograms.
A hydrogen atom has a diameter of approximately 1.06 x 10-10, as defined by the diameter of the spherical electron cloud around the nucleus. The hydrogen nucleus has a diameter of approximately 2.40 x 10-15 m. (a) For a scale model, represent the diameter of the hydrogen atom by the length of an American football field (100 yd = 300 ft), and determine the diameter of the nucleus in millimeters. (b) The atom is how many times larger in volume than its nucleus?
The nearest stars to the Sun are in the Alpha Centauri multiple-star system, about 4.0 x 1013 km away. If the Sun, with a diameter of 1.4 x 109 m, and Alpha Centauri a are both represented by cherry pits 7.0 mm in diameter, how far apart should the pits be placed to represent the Sun and its neighbor to scale?
The diameter of our disk-shaped galaxy, the Milky Way, is about 1.0 x 105 light-years (Ly). The distance to Messier 31, which is Andromeda, the spiral galaxy nearest to the Milky Way, is about 2.0 million Ly. If a scale model represents the Milky Way and Andromeda galaxies as dinner plates 25 cm in diameter, determine the distance between the two plates.
The mean radius of the Earth is 6.37 x 106 m, and that of the Moon is 1.74 x 108 cm. From these data calculate (a) The ratio of the Earth’s surface area to that of the Moon and (b) The ratio of the Earth’s volume to that of the Moon. Recall that the surface area of a sphere is 4πy2 and the volume of a sphere is 4/3 πr3
One cubic meter (1.00 m3) of aluminum has a mass of 2.70 x 103 kg, and 1.00 m3 of iron has a mass of 7.86 x 103 kg. Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 2.00 cm on an equal-arm balance.
Let & Al represent the density of aluminum and PFe that of iron. Find the radius of a solid aluminum sphere that balances a solid iron sphere of radius r Fe on an equal-arm balance.
Estimate the number of Ping-Pong balls that would fit into a typical-size room (without being crushed). In your solution state the quantities you measure or estimate and the values you take for them.
An automobile tire is rated to last for 50 000 miles. To an order of magnitude, through how many revolutions will it turn? In your solution state the quantities you measure or estimate and the values you take for them.
Grass grows densely everywhere on a quarter-acre plot of land. What is the order of magnitude of the number of blades of grass on this plot? Explain your reasoning. Note that 1 acre = 43 560 ft2.
Approximately how many raindrops fall on a one-acre lot during a one-inch rainfall? Explain your reasoning.
Compute the order of magnitude of the mass of a bathtub half full of water. Compute the order of magnitude of the mass of a bathtub half full of pennies. In your solution list the quantities you take as data and the value you measure or estimate for each.
Soft drinks are commonly sold in aluminum containers. To an order of magnitude, how many such containers are thrown away or recycled each year by U.S. consumers?
To an order of magnitude, how many piano tuners are in New York City? The physicist Enrico Fermi was famous for asking questions like this on oral Ph.D. qualifying examinations. His own facility in making order-of-magnitude calculations is exemplified in Problem 45.48.
A rectangular plate has a length of (21.3 ± 0.2) cm and a width of (9.8 ± 0.1) cm. Calculate the area of the plate, including its uncertainty.
The radius of a circle is measured to be (10.5 ± 0.2) m. Calculate the (a) Area and (b) Circumference of the circle and give the uncertainty in each value.
How many significant figures are in the following numbers? (a) 78.9 ± 0.2 (d) 0.005 3. (c) 2.46 X 10-6 (b) 3.788 X 109
The radius of a solid sphere is measured to be (6.50 ± 0.20) cm, and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.
Carry out the following arithmetic operations: (a) The sum of the measured values 756, 37.2, 0.83, and 2.5; (b) The product 0.003 2 X 356.3; (c) The product 5.620 X π.
The tropical year, the time from vernal equinox to the next vernal equinox, is the basis for our calendar. It contains 365.242199 days. Find the number of seconds in a tropical year.
A farmer measures the distance around a rectangular field. The length of the long sides of the rectangle is found to be 38.44 m, and the length of the short sides is found to be 19.5 m. What is the total distance around the field?
A sidewalk is to be constructed around a swimming pool that measures (10.0 ± 0.1) m by (17.0 ± 0.1) m. If the sidewalk is to measure (1.00 ± 0.01) m wide by (9.0 ± 0.1) cm thick, what volume of concrete is needed, and what is the approximate uncertainty of this volume?
In a situation where data are known to three significant digits, we write 6.379 m = 6.38 m and 6.374 m = 6.37m. When a number ends in 5, we arbitrarily choose to write 6.375 m = 6.38m. We could equally well write 6.375 m = 6.37 m, “rounding down” instead of “rounding up,” because we would change the number 6.375 by equal increments in both cases. Now consider an order-of-magnitude estimate, in which we consider factors rather than increments. We write 500 m ~ 103 m because 500 differs from 100 by a factor of 5 while it differs from 1 000 by only a factor of 2. We write 437 m ~ 103 m and 305 m ~ 102 m. What distance differs from 100 m and from 1 000 m by equal factors, so that we could equally well choose to represent its order of magnitude either as ~102 m or as ~103 m?
For many electronic applications, such as in computer chips, it is desirable to make components as small as possible to keep the temperature of the components low and to increase the speed of the device. Thin metallic coatings (films) can be used instead of wires to make electrical connections. Gold is especially useful because it does not oxidize readily. Its atomic mass is 197 u. A gold film can be no thinner than the size of a gold atom. Calculate the minimum coating thickness, assuming that a gold atom occupies a cubical volume in the film that is equal to the volume it occupies in a large piece of metal. This geometric model yields a result of the correct order of magnitude.
The basic function of the carburetor of an automobile is to “atomize” the gasoline and mix it with air to promote rapid combustion. As an example, assume that 30.0 cm3 of gasoline is atomized into N spherical droplets, each with a radius of 2.00 X10-5 m. What is the total surface area of these N spherical droplets?
The consumption of natural gas by a company satisfies the empirical equation V = 1.50t + 0.008 00t 2, where V is the volume in millions of cubic feet and t the time in months. Express this equation in units of cubic feet and seconds. Assign proper units to the coefficients. Assume a month is equal to 30.0 days.
In physics it is important to use mathematical approximations. Demonstrate that for small angles (<20°) tan a ≈ sin a ≈ a = πa` /180° Where, is in radians and a` is in degrees. Use a calculator to find the largest angle for which tan, may be approximated by sin, if the error is to be less than 10.0%.
A high fountain of water is located at the center of a circular pool as in Figure P1.61. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 15.0 m. Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation of the top of the fountain to be 55.0°. How high is the fountain?
Collectible coins are sometimes plated with gold to enhance their beauty and value. Consider a commemorative quarter-dollar advertised for sale at $4.98. It has a diameter of 24.1mm, a thickness of 1.78 mm, and is completely covered with a layer of pure gold 0.180 µ m thick. The volume of the plating is equal to the thickness of the layer times the area to which it is applied. The patterns on the faces of the coin and the grooves on its edge have a negligible effect on its area. Assume that the price of gold is $10.0 per gram. Find the cost of the gold added to the coin. Does the cost of the gold significantly enhance the value of the coin?
There are nearly π X 107 s in one year. Find the percentage error in this approximation, where “percentage error’’ is defined as Percentage error = | assumed value – true value | x 100 % True value
Assume that an object covers an area A and has a uniform height h. If its cross-sectional area is uniform over its height, then its volume is given by V = Ah. (a) Show that V = Ah is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form V = Ah, identifying A in each case. (Note that A, sometimes called the “footprint” of the object, can have any shape and the height can be replaced by average thickness in general.)
A child loves to watch as you fill a transparent plastic bottle with shampoo. Every horizontal cross-section is a circle, but the diameters of the circles have different values, so that the bottle is much wider in some places than others. You pour in bright green shampoo with constant volume flow rate 16.5 cm3/s. At what rate is its level in the bottle rising (a) At a point where the diameter of the bottle is 6.30 cm and (b) At a point where the diameter is 1.35 cm?
One cubic centimeter of water has a mass of 1.00 x 10-3 kg. (a) Determine the mass of 1.00 m3 of water. (b) Biological substances are 98% water. Assume that they have the same density as water to estimate the masses of a cell that has a diameter of 1.0 µm, a human kidney, and a fly. Model the kidney as a sphere with a radius of 4.0 cm and the fly as a cylinder 4.0 mm long and 2.0 mm in diameter
Assume there are 100 million passenger cars in the United States and that the average fuel consumption is 20 mi/gal of Gasoline. If the average distance traveled by each car is 10 000 mi/yr, how much gasoline would be saved per year if average fuel consumption could be increased to 25 mi/gal?
A creature moves at a speed of 5.00 furlongs per fortnight (not a very common unit of speed). Given that 1 furlong = 220 yards and 1 fortnight = 14 days, determine the speed of the creature in m/s. What kind of creature do you think it might be?
The distance from the Sun to the nearest star is about 4 x 1016 m. The Milky Way galaxy is roughly a disk of diameter ~1021 m and thickness ~1019. Find the order of magnitude of the number of stars in the Milky Way. Assume the distance between the Sun and our nearest neighbor is typical.
The position of a pine wood derby car was observed at various times; the results are summarized in the following table. Find the average velocity of the car for
(a) The first second,
(b) The last 3 s, and
(c) The entire period of observation.
t(s) 0 1.0 2.0 3.0 4.0 5.0
x(m) 0 2.3 9.2 20.7 36.8 57.5
(a) Sand dunes in a desert move over time as sand is swept up the windward side to settle in the lee side. Such “walking” dunes have been known to walk 20 feet in a year and can travel as much as 100 feet per year in particularly windy times. Calculate the average speed in each case in m/s.
(b) Fingernails grow at the rate of drifting continents, on the order of 10 mm/yr. Approximately how long did it take for North America to separate from Europe, a distance of about 3 000 mi?
The position versus time for a certain particle moving along the x axis is shown in Figure P2.3. Find the average velocity in the time intervals
(a) 0 to 2 s,
(b) 0 to 4 s,
(c) 2 s to 4 s,
(d) 4 s to 7 s,
(e) 0 to 8 s
A particle moves according to the equation x = 10 t 2 where x is in meters and t is in seconds
(a) Find the average velocity for the time interval from 2.00 to 3.00 s.
(b) Find the average velocity for the time interval from 2.00 to 2.10 s.
A person walks first at a constant speed of 5.00 m/s along a straight line from point A to point B and then back along the line from B to A at a constant speed of 3.00 m/s.
What is
(a) Her average speed over the entire trip?
(b) Her average velocity over the entire trip?
The position of a particle moving along the x axis varies in time according to the expression x = 3t2, where x is in meters and t is in seconds. Evaluate its position
(a) At t = 3.00 s and
(b) At 3.00 s + ∆t
(c) Evaluate the limit of ∆x/ ∆t as ∆t approaches zero, to find the velocity at t = 3.00 s
A position-time graph for a particle moving along the x axis is shown in Figure P2.7.
(a) Find the average velocity in the time interval t = 1.50 s to t = 4.00 s
(b) Determine the instantaneous velocity at t = 2.00 s by measuring the slope of the tangent line shown in the graph
(c) At what value of t is the velocity zero?
(a) Use the data in Problem 1 to construct a smooth graph of position versus time.
(b) By constructing tangents to the x (t) curve, find the instantaneous velocity of the car at several instants.
(c) Plot the instantaneous velocity versus time and, from this, determine the average acceleration of the car.
(d) What was the initial velocity of the car?
Find the instantaneous velocity of the particle described in Figure P2.3 at the following times:
(a) t = 1.0 s,
(b) t = 3.0 s
(c) t = 4.5 s, and
(d) t = 7.5 s
A hare and a tortoise compete in a race over a course 1.00 km long. The tortoise crawls straight and steadily at its maximum speed of 0.200 m/s toward the finish line. The hare runs at its maximum speed of 8.00 m/s toward the goal for 0.800 km and then stops to tease the tortoise. How close to the goal can the hare let the tortoise approach before resuming the race, which the tortoise wins in a photo finish? Assume that, when moving, both animals move steadily at their respective maximum speeds.
The data in the following table represent measurements of the masses and dimensions of solid cylinders of aluminum, copper, brass, tin, and iron. Use these data to calculate the densities of these substances. Compare your results for aluminum, copper, and iron with those given in Table 1.5.
A 50.0-g superb all traveling at 25.0 m/s bounces off a brick wall and rebounds at 22.0 m/s. A high-speed camera records this event. If the ball is in contact with the wall for 3.50 ms, what is the magnitude of the average acceleration of the ball during this time interval? (Note: 1 ms = 10-3 s)
A particle starts from rest and accelerates as shown in Figure P2.12. Determine
(a) The particle’s speed at t = 10.0 s and at t = 20.0 s, and
(b) The distance traveled in the first 20.0 s.
Secretariat won the Kentucky Derby with times for successive quarter-mile segments of 25.2 s, 24.0 s, 23.8, and 23.0 s.
(a) Find his average speed during each quarter-mile segment.
(b) Assuming that Secretariat’s instantaneous speed at the finish line was the same as the average speed during the final quarter mile; find his average acceleration for the entire race. (Horses in the Derby start from rest.)
A velocity–time graph for an object moving along the x axis is shown in Figure P2.14. ]
(a) Plot a graph of the acceleration versus time.
(b) Determine the average acceleration of the object in the time intervals t = 5.00 s to t = 15.0 s and t = 0 to t = 20.0 s
A particle moves along the x axis according to the equation x = 2.00 + 3.00t - 1.00t 2, where x is in meters and t is in seconds. At t = 3.00 s, find
(a) The position of the particle,
(b) Its velocity, and
(c) Its acceleration.
An object moves along the x axis according to the equation x (t) = (3.00t2 - 2.00t + 3.00) m. Determine
(a) The average speed between t = 2.00 s and t = 3.00 s,
(b) The instantaneous speed at t = 2.00 s and at t = 3.00 s,
(c) The average acceleration between t = 2.00 s and t = 3.00 s, and
(d) The instantaneous acceleration at t = 2.00 s and t = 3.00 s
Figure P2.17 shows a graph of vx versus t for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line.
(a) Find the average acceleration for the time interval t = 0 to t = 6.00 s.
(b) Estimate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant.
(c) When is the acceleration zero?
(d) Estimate the maximum negative value of the acceleration and the time at which it occurs.
Draw motion diagrams for
(a) An object moving to the right at constant speed,
(b) An object moving to the right and speeding up at a constant rate,
(c) An object moving to the right and slowing down at a constant rate,
(d) An object moving to the left and speeding up at a constant rate, and
(e) An object moving to the left and slowing down at a constant rate. (f) How would your drawings change if the changes in speed were not uniform; that is, if the speed were not changing at a constant rate?
Jules Verne in 1865 suggested sending people to the Moon by firing a space capsule from 220-m-long cannon with a launch speed of 10.97 km/s. What would have been the unrealistically large acceleration experienced by the space travelers during launch? Compare your answer with the free-fall acceleration 9.80 m/s2.
A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final speed of 2.80 m/s.
(a) Find its original speed.
(b) Find its acceleration.
An object moving with uniform acceleration has a velocity of 12.0 cm/s in the positive x direction when its x coordinate is 3.00 cm. If its x coordinate 2.00 s later is -5.00 cm, what is its acceleration?
A 745i BMW car can brake to a stop in a distance of 121 ft. from a speed of 60.0 mi/h. To brake to a stop from a speed of 80.0 mi/h requires a stopping distance of 211 ft. What is the average braking acceleration for?
(a) 60 mi/h to rest,
(b) 80 mi/h to rest,
(c) 80 mi/h to 60 mi/h? Express the answers in mi/h/s and in m/s2
A speedboat moving at 30.0 m/s approaches a no-wake buoy marker 100m ahead. The pilot slows the boat with a constant acceleration of - 3.50 m/s2 by reducing the throttle.
(a) How long does it take the boat to reach the buoy?
(b) What is the velocity of the boat when it reaches the buoy?
Figure P2.24 represents part of the performance data of a car owned by a proud physics student.
(a) Calculate from the graph the total distance traveled.
(b) What distance does the car travel between the times t = 10 s and t = 40 s?
(c) Draw a graph of its acceleration versus time between t = 0 and t= 50 s.
(d) Write an equation for x as a function of time for each phase of the motion, represented by (i) 0a, (ii) ab, (iii) bc.
(e) What is the average velocity of the car between t = 0 and t = 50 s?
A particle moves along the x axis. Its position is given by the equation x = 2 x 3t - 4t2 with x in meters and t in seconds. Determine
(a) Its position when it changes direction and
(b) Its velocity when it returns to the position it had at t = 0.
In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straightaway at 71.5 m/s. The driver of the Thunderbird realizes he must make a pit stop, and he smoothly slows to a stop over a distance of 250 m. He spends 5.00 s in the pit and then accelerates out, reaching his previous speed of 71.5 m/s after a distance of 350 m. At this point, how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?
A jet plane lands with a speed of 100 m/s and can accelerate at a maximum rate of - 5.00 m/s2 as it comes to rest.
(a) From the instant the plane touches the runway, what is the minimum time interval needed before it can come to rest?
(b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?
A car is approaching a hill at 30.0 m/s when its engine suddenly fails just at the bottom of the hill. The car moves with a constant acceleration of -2.00 m/s2 while coasting up the hill.
(a) Write equations for the position along the slope and for the velocity as functions of time, taking x = 0 at the bottom of the hill, where vi=30.0 m/s.
(b) Determine the maximum distance the car rolls up the hill.
Help! One of our equations is missing! We describe constant acceleration motion with the variables and parameters vxi, vxf, ax, t, and xf - xi. Of the equations in Table 2.2, the first does not involve xf - xi . The second does not contain ax; the third omits vxf and the last leaves out t. So to complete the set there should be an equation not involving vxi. Derive it from the others. Use it to solve Problem 29 in one step. Discuss.
For many years Colonel John P. Step, USAF, held the world’s land speed record. On March 19, 1954, he rode a rocket-propelled sled that moved down a track at a speed of 632 mi/h. He and the sled were safely brought to rest in 1.40 s (Fig. P2.31). Determine
(a) The negative acceleration he experienced and
(b) The distance he traveled during this negative acceleration.
A truck on a straight road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 20.0 m/s. Then the truck travels for 20.0 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.00 s.
(a) How long is the truck in motion?
(b) What is the average velocity of the truck for the motion described?
An electron in a cathode ray tube (CRT) accelerates from 2.00 x 104 m/s to 6.00 x 106 m/s over 1.50 cm.
(a) How long does the electron take to travel this 1.50 cm?
(b) What is its acceleration?
In a 100-m linear accelerator, an electron is accelerated to 1.00% of the speed of light in 40.0m before it coasts for 60.0m to a target.
(a) What is the electron’s acceleration during the first 40.0 m?
(b) How long does the total flight take?
Within a complex machine such as a robotic assembly line, suppose that one particular part glides along a straight track. A control system measures the average velocity of the part during each successive interval of time ∆t0 =t0 - 0, compares it with the value vc it should be, and switches a servo motor on and off to give the part a correcting pulse of acceleration. The pulse consists of a constant acceleration am applied for time interval ∆tm=tm-0 within the next control time interval ∆t0. As shown in Fig. P2.35, the part may be modeled as having zero acceleration when the motor is off (between tm and t0). A computer in the control system chooses the size of the acceleration so that the final velocity of the part will have the correct value vc . Assume the part is initially at rest and is to have instantaneous velocity vc at time t0.
(a) Find the required value of am in terms of vc and tm.
(b) Show that the displacement ∆x of the part during the time interval ∆t0 is given by ∆x=vc (t0 - 0.5tm). For specified values of vc and t0,
(c) What is the minimum displacement of the part?
(d) What is the maximum displacement of the part?
(e) Are both the minimum and maximum displacements physically attainable?
A glider on an air track carries a flag of length l through a stationary photo gate, which measures the time interval ∆td during which the flag blocks a beam of infrared light passing across the photo gate. The ratio vd l / ∆td is the average velocity of the glider over this part of its motion. Suppose the glider moves with constant acceleration.
(a) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photo gate in space.
(b) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photo gate in time.
A ball starts from rest and accelerates at 0.500 m/s2 while moving down an inclined plane 9.00 m long. When it reaches the bottom, the ball rolls up another plane, where, after moving 15.0 m, it comes to rest.
(a) What is the speed of the ball at the bottom of the first plane?
(b) How long does it take to roll down the first plane?
(c) What is the acceleration along the second plane?
(d) What is the ball’s speed 8.00 m along the second plane?
Speedy Sue, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at - 2.00 m/s 2 because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sue’s car and the van
Solve Example 2.8, “Watch out for the Speed Limit!” by a graphical method. On the same graph plot position versus time for the car and the police officer, from the intersection of the two curves read the time at which the trooper overtakes the car
A golf ball is released from rest from the top of a very tall building. Neglecting air resistance, calculate
(a) The position and
(b) The velocity of the ball after 1.00, 2.00, and 3.00 s
Every morning at seven o’clock there’s twenty terriers drilling on the rock. The boss comes around and he says, “Keep still and bear down heavy on the cast-iron drill and drill, ye terriers, drill.” And drills, ye terriers, drill its work all day for sugar in your tea down beyond the railway. And drill, ye terriers, drill .The foreman’s name was John McAnn. By God, he was a blamed mean man. One day a premature blast went off And a mile in the air went big Jim Goff. And drill ... Then when next payday came around Jim Goff a dollar short was found. When he asked what for, came this reply: “You were docked for the time you were up in the sky.” And drill... —American folksong What was Goff’s hourly wage? State the assumptions you make in computing it.
A ball is thrown directly downward, with an initial speed of 8.00 m/s, from a height of 30.0 m. After what time interval does the ball strike the ground?
A student throws a set of keys vertically upward to her sorority sister, who is in a window 4.00 m above. The keys are caught 1.50 s later by the sister’s outstretched hand.
(a) With what initial velocity were the keys thrown?
(b) What was the velocity of the keys just before they were caught?
Emily challenges her friend David to catch a dollar bill as follows. She holds the bill vertically, as in Figure P2.44, with the center of the bill “between David” is index finger and thumb. David must catch the bill after Emily releases it without moving his hand downward. If his reaction time is 0.2 s, will he succeed? Explain your reasoning.
In Mostar, Bosnia, the ultimate test of a young man’s courage once was to jump off a 400-year-old bridge (now destroyed) into the River Neretva, 23.0 m below the bridge.
(a) How long did the jump last?
(b) How fast was the diver traveling upon impact with the water?
(c) If the speed of sound in air is 340 m/s, how long after the diver took off did a spectator on the bridge hear the splash?
A ball is dropped from rest from a height h above the ground. Another ball is thrown vertically upwards from the ground at the instant the first ball is released. Determine the speed of the second ball if the two balls are to meet at a height h/2 above the ground.
A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes 3.00 s for the ball to reach its maximum height. Find
(a) Its initial velocity and
(b) The height it reaches.
It is possible to shoot an arrow at a speed as high as100 m/s.
(a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up?
(b) How long would the arrow be in the air?
A daring ranch hand sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 10.0 m/s, and the distance from the limb to the level of the saddle is 3.00 m. (a) What must be the horizontal distance between the saddle and limb when the ranch hand makes his move? (b) How long is he in the air?
A woman is reported to have fallen 144 ft from the 17th floor of a building, landing on a metal ventilator box, which she crushed to a depth of 18.0 in. She suffered only minor injuries. Neglecting air resistance, calculate (a) The speed of the woman just before she collided with the ventilator, (b) Her average acceleration while in contact with the box, and (c) The time it took to crush the box.
The height of a helicopter above the ground is given by h = 3.00t 3, where h is in meters and t is in seconds. After 2.00 s, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?
A freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall?
Automotive engineers refer to the time rate of change of acceleration as the “jerk.” If an object moves in one dimension such that its jerk J is constant,
(a) Determine expressions for its acceleration ax (t), velocity vx (t), and position x (t), given that its initial acceleration, velocity, and position are axi , vxi , and xi , respectively.
(b) Show that ax2 = axi2 + 2J (vx – vxi).
A student drives a moped along a straight road as described by the velocity-versus-time graph in Figure P2.54. Sketch this graph in the middle of a sheet of graph paper.
(a) Directly above your graph, sketch a graph of the position versus time, aligning the time coordinates of the two graphs.
(b) Sketch a graph of the acceleration versus time directly below the vx-t graph, again aligning the time coordinates. On each graph, show the numerical values of x and ax for all points of inflection.
(c) What is the acceleration? at t = 6 s?
(d) Find the position (relative to the starting point) at t = 6 s.
(e) What is the moped’s final position at t = 9 s?
The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by v = (- 5.00 x 107)t 2 + (3.00 x 105)t, where v is in meters per second and t is in seconds. The acceleration of the bullet just as it leaves the barrel is zero.
(a) Determine the acceleration and position of the bullet as a function of time when the bullet is in the barrel.
(b) Determine the length of time the bullet is accelerated.
(c) Find the speed at which the bullet leaves the barrel.
(d) What is the length of the barrel?
The acceleration of a marble in a certain fluid is proportional to the speed of the marble squared, and is given (in SI units) by a = -3.00v 2 for v > 0. If the marble enters this fluid with a speed of 1.50 m/s, how long will it take before the marble’s speed is reduced to half of its initial value?
A car has an initial velocity v0 when the driver sees an obstacle in the road in front of him. His reaction time is ∆tr, and the braking acceleration of the car is a. Show that the total stopping distance is sstop = vo ∆tr – vo2/2a. Remember that a is a negative number.
The yellow caution light on a traffic signal should stay on long enough to allow a driver to either pass through the intersection or safely stop before reaching the intersection. A car can stop if its distance from the intersection is greater than the stopping distance found in the previous problem. If the car is less than this stopping distance from the intersection, the yellow light should stay on long enough to allow the car to pass entirely through the intersection.
(a) Show that the yellow light should stay on for a time interval ∆ttight = ∆t r – (vo / 2a) + (si / vo) Where ∆tr is the driver’s reaction time, v0 is the velocity of the car approaching the light at the speed limit, a is the braking acceleration, and si is the width of the intersection.
(b) As city traffic planner, you expect cars to approach an intersection 16.0 m wide with a speed of 60.0 km/h. Be cautious and assume a reaction time of 1.10 s to allow for a driver’s indecision. Find the length of time the yellow light should remain on. Use a braking acceleration of - 2.00 m/s2.
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