Chapter 3

(II) You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 4.0 s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?

You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes \(4.0 \mathrm{~s}\) for the dart to land back at the barrel. What is the maximum horizontal range of your gun?

(II) A baseball is hit with a specd of 27.0 \(\mathrm{m} / \mathrm{s}\) at an angle of \(45.0^{\circ} .\) It lands on the flat roof of a \(13.0-\mathrm{m}\) -tall nearby building. If the ball was hit when it was 1.0 \(\mathrm{m}\) above the ground, what horizontal distance does it travel before it lands on the building?

(II) A baseball is hit with a speed of 27.0 \(\mathrm{m} / \mathrm{s}\) at an angle of \(45.0^{\circ} .\) It lands on the flat roof of a \(13.0-\mathrm{m}\) -tall nearby building. If the ball was hit when it was 1.0 \(\mathrm{m}\) above the ground, what horizontal distance does it travel before it lands on the building?

A baseball is hit with a speed of \(27.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(45.0^{\circ} .\) It lands on the flat roof of a 13.0 -m-tall nearby building. If the ball was hit when it was \(1.0 \mathrm{~m}\) above the ground, what horizontal distance does it travel before it lands on the building?

A grasshopper hops down a level road. On each hop, the grasshopper launches itself at angle \(\theta_{0}=45^{\circ}\) and achieves a range \(R=1.0 \mathrm{~m}\). What is the average horizontal speed of the grasshopper as it progresses down the road? Assume that the time spent on the ground between hops is negligible.

Extreme-sports enthusiasts have been known to jump off the top of El Capitan, a sheer granite cliff of height \(910 \mathrm{~m}\) in Yosemite National Park. Assume a jumper runs horizontally off the top of El Capitan with speed \(5.0 \mathrm{~m} / \mathrm{s}\) and enjoys a freefall until she is \(150 \mathrm{~m}\) above the valley floor, at which time she opens her parachute (Fig. \(3-41\) ). (a) How long is the jumper in freefall? Ignore air resistance. (b) It is important to be as far away from the cliff as possible before opening the parachute. How far from the cliff is this jumper when she opens her chute?

(II) Extreme-sports enthusiasts have been known to jump off the top of El Capitan, a shecr granite cliff of height 910 \(\mathrm{m}\) in Yosemite National Park. Assume a jumper runs horizontally off the top of EI Capitan with speed 5.0 \(\mathrm{m} / \mathrm{s}\) and enjoys a freefall until she is 150 \(\mathrm{m}\) above the valley floor, at which time she opens her parachute (Fig. 41\()\) .(a) How long is the jumper in frecfall? Ignore air resistance. \((b)\) It is important to be as far away from the cliff as possible before opening the parachute. How far from the cliff is this jumper when she opens her chute?

The pilot of an airplane traveling \(170 \mathrm{~km} / \mathrm{h}\) wants to drop supplies to flood victims isolated on a patch of land \(150 \mathrm{~m}\) below. The supplies should be dropped how many seconds before the plane is directly overhead?

\((a)\) A long jumper leaves the ground at \(45^{\circ}\) above the horizontal and lands \(8.0 \mathrm{~m}\) away. What is her "takeoff' speed \(v_{0} ?(b)\) Now she is out on a hike and comes to the left bank of a river. There is no bridge and the right bank is \(10.0 \mathrm{~m}\) away horizontally and \(2.5 \mathrm{~m},\) vertically below. If she long jumps from the edge of the left bank at \(45^{\circ}\) with the speed calculated in \((a),\) how long, or short, of the opposite bank will she land (Fig. \(3-43\) )?

(11) (a) A long jumper leaves the ground at \(45^{\circ}\) above the horizontal and lands 8.0 \(\mathrm{m}\) away. What is her "takeoff" spced \(v_{0} ?(b)\) Now she is out on a hike and comes to the left bank of a river. There is no bridge and the right bank is 10.0 \(\mathrm{m}\) away horizontally and \(2.5 \mathrm{m},\) vertically below. If she long jumps from the edge of the left bank at \(45^{\circ}\) with the speed calculated in \((a),\) how long, or short, of the opposite bank will she land (Fig. 43\() ?\)

A high diver leaves the end of a 5.0 -m-high diving board and strikes the water \(1.3 \mathrm{~s}\) later, \(3.0 \mathrm{~m}\) beyond the end of the board. Considering the diver as a particle, determine (a) her initial velocity, \(\overrightarrow{\mathbf{v}}_{0} ;(b)\) the maximum height reached; and \((c)\) the velocity \(\overrightarrow{\mathbf{v}}_{\mathrm{f}}\) with which she enters the water.

(II) A high diver leaves the end of a 5.0 -m-high diving board and strikes the water 1.3 s later, 3.0 \(\mathrm{m}\) beyond the cnd of the board. Considering the diver as a particle, determine (a) her initial velocity, \(\vec{\mathbf{v}}_{0} ;(b)\) the maximum height reached: and \((c)\) the velocity \(\overline{\mathbf{v}}_{\mathrm{f}}\) with which she enters the water.

A ball is thrown horizontally from the top of a cliff with initial speed \(v_{0}\) (at \(t=0\) ). At any moment, its direction of motion makes an angle \(\theta\) to the horizontal (Fig. \(3-47\) ). Derive a formula for \(\theta\) as a function of time, \(t,\) as the ball follows a projectile's path.

(II) A ball is thrown horizontally from the top of a cliff with initial speed \(v_{0}\) (at \(t=0 )\) . At any moment, its direction of motion makes an angle \(\theta\) to the horizontal (Fig. 47\()\) . Derive a formula for \(\theta\) as a function of time, \(t,\) as the ball follows a projectile's path.

(II) At what projection angle will the range of a projectile cqual its maximum height?

A projectile is fired with an initial speed of \(46.6 \mathrm{~m} / \mathrm{s}\) at an angle of \(42.2^{\circ}\) above the horizontal on a long flat firing range. Determine \((a)\) the maximum height reached by the projectile, \((b)\) the total time in the air, \((c)\) the total horizontal distance covered (that is, the range), and \((d)\) the velocity of the projectile \(1.50 \mathrm{~s}\) after firing.

(1I) A projectile is fired with an initial speed of 46.6 \(\mathrm{m} / \mathrm{s}\) at an angle of \(42.2^{\circ}\) above the horizontal on a long flat firing range. Determine \((a)\) the maximum height reached by the projectile, (b) the total time in the air, \((c)\) the total horizontal distance covered (that is, the range), and (d) the velocity of the projectile 1.50 s after firing.

(II) An athlete executing a long jump leaves the ground at a \(27.0^{\circ}\) angle and lands 7.80 \(\mathrm{m}\) away. (a) What was the takeoff spced? (b) If this speed were increased by just 5.0\(\%\) , how much longer would the jump be?

(III) A person stands at the base of a hill that is a straight incline making an angle \(\phi\) with the horizontal (Fig. 48). For a given initial spced \(v_{0},\) at what angle \(\theta\) (to the horizontal) should objects be thrown so that the distance \(d\) they land up the hill is as large as possible?

Derive a formula for the horizontal range \(R,\) of a projectile when it lands at a height \(h\) above its initial point. (For \(h<0\), it lands a distance \(-h\) below the starting point.) Assume it is projected at an angle \(\theta_{0}\) with initial speed \(v_{0}\).

(III) Derive a formula for the horizontal range \(R,\) of a projectile when it lands at a height \(h\) above its initial point. (For \(h<0,\) it lands a distance \(-h\) below the starting point.) Assume it is projected at an angle \(\theta_{0}\) with initial speed \(v_{0}\) .

A person going for a morning jog on the deck of a cruise ship is running toward the bow (front) of the ship at \(2.0 \mathrm{~m} / \mathrm{s}\) while the ship is moving ahead at \(8.5 \mathrm{~m} / \mathrm{s}\). What is the velocity of the jogger relative to the water? Later, the jogger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now?

(1) A person going for a morning jog on the deck of a cruise ship is running toward the bow (front) of the ship at 2.0 \(\mathrm{m} / \mathrm{s}\) while the ship is moving ahead at 8.5 \(\mathrm{m} / \mathrm{s}\) . What is the velocity of the jogger relative to the water? Later, the joger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now?

Huck Finn walks at a speed of \(0.70 \mathrm{~m} / \mathrm{s}\) across his raft (that is, he walks perpendicular to the raft's motion relative to the shore). The raft is traveling down the Mississippi River at a speed of \(1.50 \mathrm{~m} / \mathrm{s}\) relative to the river bank (Fig. \(3-49\) ). What is Huck's velocity (speed and direction) relative to the river bank?

Two planes approach each other head-on. Each has a speed of \(780 \mathrm{~km} / \mathrm{h}\), and they spot each other when they are initially \(12.0 \mathrm{~km}\) apart. How much time do the pilots have to take evasive action?

(11) Two planes approach cach other head-on. Each has a speed of 780 \(\mathrm{km} / \mathrm{h}\) , and they spot cach other when they are initially 12.0 \(\mathrm{km}\) apart. How much time do the pilots have to take evasive action?

A child, who is \(45 \mathrm{~m}\) from the bank of a river, is being carried helplessly downstream by the river's swift current of \(1.0 \mathrm{~m} / \mathrm{s} .\) As the child passes a lifeguard on the river's bank, the lifeguard starts swimming in a straight line until she reaches the child at a point downstream (Fig. \(3-50\) ). If the lifeguard can swim at a speed of \(2.0 \mathrm{~m} / \mathrm{s}\) relative to the water, how long does it take her to reach the child? How far downstream does the lifeguard intercept the child?

(II) A child, who is 45 \(\mathrm{m}\) from the bank of a river, is being carricd helplessly downstream by the river's swift current of 1.0 \(\mathrm{m} / \mathrm{s} .\) As the child passes a lifeguard on the river's bank, the lifeguard starts swimming in a straight line untill she reaches the child at a point downstream (Fig. 50\()\) . If the lifeguard can swim at a speed of 2.0 \(\mathrm{m} / \mathrm{s}\) relative to the water, how long does it take her to reach the child? How far downstream does the lifeguard intercept the child?

A person in the passenger basket of a hot-air balloon throws a ball horizontally outward from the basket with speed \(10.0 \mathrm{~m} / \mathrm{s}\) (Fig. \(3-52\) ). What initial velocity (magnitude and direction) does the ball have relative to a person standing on the ground \((a)\) if the hot-air balloon is rising at \(5.0 \mathrm{~m} / \mathrm{s}\) relative to the ground during this throw, (b) if the hot-air balloon is descending at \(5.0 \mathrm{~m} / \mathrm{s}\) relative to the ground.

(II) A person in the passenger basket of a hot-air balloon throws a ball horizontally outward from the basket with spced 10.0 \(\mathrm{m} / \mathrm{s}\) (Fig. 52\()\) . What initial velocity (magnitude and direction) does the ball have relative to a person standing on the ground (a) if the hot-air balloon is rising at 5.0 \(\mathrm{m} / \mathrm{s}\) relative to the ground during this throw, (b) if the hot-air balloon is descending at 5.0 \(\mathrm{m} / \mathrm{s}\) relative to the ground.

An airplane is heading due south at a speed of \(580 \mathrm{~km} / \mathrm{h}\). If a wind begins blowing from the southwest at a speed of \(90.0 \mathrm{~km} / \mathrm{h}\) (average), calculate (a) the velocity (magnitude and direction) of the plane, relative to the ground, and (b) how far from its intended position it will be after \(11.0 \mathrm{~min}\) if the pilot takes no corrective action. [Hint: First draw a diagram.

(II) An airplane is heading due south at a speed of 580 \(\mathrm{km} / \mathrm{h}\) . If a wind begins blowing from the southwest at a speed of 90.0 \(\mathrm{km} / \mathrm{h}\) (average), calculate \((a)\) the velocity (magnitude and dircction) of the plane, relative to the ground, and (b) how far from its intended position it will be after 11.0 \(\mathrm{min}\) if the pilot takes no corrective action.

Two cars approach a street corner at right angles to each other (see Fig. 3-35). Car 1 travels at \(35 \mathrm{~km} / \mathrm{h}\) and car 2 at \(45 \mathrm{~km} / \mathrm{h}\). What is the relative velocity of car 1 as seen by car 2 ? What is the velocity of car 2 relative to car \(1 ?\)

(II) Two cars approach a street corner at right angles to each other (sce Fig. \(35 ) .\) Car 1 travels at 35 \(\mathrm{km} / \mathrm{h}\) and car 2 at 45 \(\mathrm{km} / \mathrm{h}\) . What is the relative velocity of car 1 as scen by car 2\(?\) What is the velocity of car 2 relative to car 1\(?\)

A swimmer is capable of swimming \(0.60 \mathrm{~m} / \mathrm{s}\) in still water. \((a)\) If she aims her body directly across a \(55-\mathrm{m}\) -wide river whose current is \(0.50 \mathrm{~m} / \mathrm{s}\), how far downstream (from a point opposite her starting point) will she land? (b) How long will it take her to reach the other side?

(1I) A swimmer is capable of swimming 0.60 \(\mathrm{m} / \mathrm{s}\) in still water. (a) If she aims her body directly across a 55 -m-wide river whose current is 0.50 \(\mathrm{m} / \mathrm{s}\) , how far downstream (from a point opposite her starting point) will she land? (b) How long will it take her to reach the other side?

(II) A motorboat whose speed in still water is 3.40 \(\mathrm{m} / \mathrm{s}\) must aim upstream at an angle of \(19.5^{\circ}\) (with respect to a line perpendicular to the shore) in order to travel dircctly across the stream. (a) What is the spced of the current? (b) What is the resultant speed of the boat with respect to the shore?

An airplane, whose air speed is \(580 \mathrm{~km} / \mathrm{h},\) is supposed to fly in a straight path \(38.0^{\circ} \mathrm{N}\) of \(\mathrm{E}\). But a steady \(72 \mathrm{~km} / \mathrm{h}\) wind is blowing from the north. In what direction should the plane head?

(III) An airplanc, whose air speed is 580 \(\mathrm{km} / \mathrm{h}\) , is supposed to fly in a straight path \(38.0^{\circ} \mathrm{N}\) of E. But a steady 72 \(\mathrm{km} / \mathrm{h}\) wind is blowing from the north. In what direction should the plane head?

A plumber steps out of his truck, walks \(66 \mathrm{~m}\) east and \(35 \mathrm{~m}\) south, and then takes an elevator \(12 \mathrm{~m}\) into the subbasement of a building where a bad leak is occurring. What is the displacement of the plumber relative to his truck? Give your answer in components; also give the magnitude and angles, with respect to the \(x\) axis, in the vertical and horizontal plane. Assume \(x\) is east, \(y\) is north, and \(z\) is up.

A light plane is headed due south with a speed relative to still air of \(185 \mathrm{~km} / \mathrm{h}\). After \(1.00 \mathrm{~h}\), the pilot notices that they have covered only \(135 \mathrm{~km}\) and their direction is not south but southeast \(\left(45.0^{\circ}\right) .\) What is the wind velocity?

A light plane is headed due south with a specd relative to still air of 185 \(\mathrm{km} / \mathrm{h}\) . After 1.00 \(\mathrm{h}\) , the pilot notices that they have covered only 135 \(\mathrm{km}\) and their direction is not south but southeast \(\left(45.0^{\circ}\right) .\) What is the wind velocity?

An Olympic long jumper is capable of jumping \(8.0 \mathrm{~m}\). Assuming his horizontal speed is \(9.1 \mathrm{~m} / \mathrm{s}\) as he leaves the ground, how long is he in the air and how high does he go? Assume that he lands standing upright-that is, the same way he left the ground.

Romeo is chucking pebbles gently up to Juliet's window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden \(8.0 \mathrm{~m}\) below her window and \(9.0 \mathrm{~m}\) from the base of the wall (Fig. 3-55). How fast are the pebbles going when they hit her window?

Romeo is chucking pebbles gently up to Julict's window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 8.0 below her window and 9.0 \(\mathrm{m}\) from the base of the wall (Fig. 55 ). How fast are the pebbles going when they hit her window?

Apollo astronauts took a "nine iron" to the Moon and hit a golf ball about \(180 \mathrm{~m}\). Assuming that the swing, launch angle, and so on, were the same as on Earth where the same astronaut could hit it only \(32 \mathrm{~m}\), estimate the acceleration due to gravity on the surface of the Moon. (We neglect air resistance in both cases, but on the Moon there is none.)

A hunter aims directly at a target (on the same level) \(68.0 \mathrm{~m}\) away. \((a)\) If the bullet leaves the gun at a speed of \(175 \mathrm{~m} / \mathrm{s}\) by how much will it miss the target? \((b)\) At what angle should the gun be aimed so the target will be hit?

When Babe Ruth hit a homer over the 8.0 -m-high rightfield fence \(98 \mathrm{~m}\) from home plate, roughly what was the minimum speed of the ball when it left the bat? Assume the ball was hit \(1.0 \mathrm{~m}\) above the ground and its path initially made a \(36^{\circ}\) angle with the ground.

The speed of a boat in still water is \(v .\) The boat is to make a round trip in a river whose current travels at speed \(u\). Derive a formula for the time needed to make a round trip of total distance \(D\) if the boat makes the round trip by moving (a) upstream and back downstream, and ( \(b\) ) directly across the river and back. We must assume \(u