Chapter 3

An airplane climbs at an angle of \(15^{\circ}\) with a horizontal component of speed of \(200 \mathrm{~km} / \mathrm{h}\) (a) What is the plane's actual speed? (b) What is the magnitude of the vertical component of its velocity?

A golf ball is hit with an initial speed of \(35 \mathrm{~m} / \mathrm{s}\) at an angle less than \(45^{\circ}\) above the horizontal. (a) The horizontal velocity component is (1) greater than, (2) equal to, (3) less than the vertical velocity component. Why? (b) If the ball is hit at an angle of \(37^{\circ},\) what are the initial horizontal and vertical velocity components?

The \(x\) - and \(y\) -components of an acceleration vector are \(3.0 \mathrm{~m} / \mathrm{s}^{2}\) and \(4.0 \mathrm{~m} / \mathrm{s}^{2},\) respectively. (a) The magnitude of the acceleration vector is (1) less than \(3.0 \mathrm{~m} / \mathrm{s}^{2}\) (2) between \(3.0 \mathrm{~m} / \mathrm{s}^{2}\) and \(4.0 \mathrm{~m} / \mathrm{s}^{2}\) (3) between \(4.0 \mathrm{~m} / \mathrm{s}^{2}\) and \(7.0 \mathrm{~m} / \mathrm{s}^{2},(4)\) equal to \(7.0 \mathrm{~m} / \mathrm{s}^{2} .\) (b) What are the magnitude and direction of the acceleration vector?

If the magnitude of a velocity vector is \(7.0 \mathrm{~m} / \mathrm{s}\) and the \(x\) -component is \(3.0 \mathrm{~m} / \mathrm{s},\) what is the \(y\) -component?

The \(x\) -component of a velocity vector that has an angle of \(37^{\circ}\) to the \(+x\) -axis has a magnitude of \(4.8 \mathrm{~m} / \mathrm{s}\) (a) What is the magnitude of the velocity? (b) What is the magnitude of the \(y\) -component of the velocity?

A student walks \(100 \mathrm{~m}\) west and \(50 \mathrm{~m}\) south. (a) To get back to the starting point, the student must walk in a general direction of (1) south of west, (2) north of east, (3) south of east, (4) north of west. (b) What displacement will bring the student back to the starting point?

A student strolls diagonally across a level rectangular campus plaza, covering the 50 -m distance in 1.0 min (vFig. 3.25). (a) If the diagonal route makes a \(37^{\circ}\) angle with the long side of the plaza, what would be the distance traveled if the student had walked halfway around the outside of the plaza instead of along the diagonal route? (b) If the student had walked the outside route in 1.0 min at a constant speed, how much time would she have spent on each side?

A ball rolls at a constant velocity of \(1.50 \mathrm{~m} / \mathrm{s}\) at an angle of \(45^{\circ}\) below the \(+x\) -axis in the fourth quadrant. If we take the ball to be at the origin at \(t=0\) what are its coordinates \((x, y) 1.65\) s later?

A ball rolling on a table has a velocity with rectangular components \(v_{x}=0.60 \mathrm{~m} / \mathrm{s}\) and \(v_{y}=0.80 \mathrm{~m} / \mathrm{s} .\) What is the displacement of the ball in an interval of \(2.5 \mathrm{~s} ?\)

A hot air balloon rises vertically with a speed of \(1.5 \mathrm{~m} / \mathrm{s}\). At the same time, there is a horizontal \(10 \mathrm{~km} / \mathrm{h}\) wind blowing. In which direction is the balloon moving?

During part of its trajectory (which lasts exactly \(1 \mathrm{~min}\) ) a missile travels at a constant speed of \(2000 \mathrm{mi} / \mathrm{h}\) while maintaining a constant orientation angle of \(20^{\circ}\) from the vertical. (a) During this phase, what is true about its velocity components: \((1) v_{y}>v_{x},\) (2) \(v_{y}=v_{x},\) or (3) \(v_{y}

At the instant a ball rolls off a rooftop it has a horizontal velocity component of \(+10.0 \mathrm{~m} / \mathrm{s}\) and a vertical component (downward) of \(15.0 \mathrm{~m} / \mathrm{s}\). (a) Determine the angle of the roof. (b) What is the ball's speed as it leaves the roof?

A particle moves at a speed of \(3.0 \mathrm{~m} / \mathrm{s}\) in the \(+x\) -direction. Upon reaching the origin, the particle receives a continuous constant acceleration of \(0.75 \mathrm{~m} / \mathrm{s}^{2}\) in the \(-y\) -direction. What is the position of the particle 4.0 s later?

At a constant speed of \(60 \mathrm{~km} / \mathrm{h}\), an automobile travels \(700 \mathrm{~m}\) along a straight highway that is inclined \(4.0^{\circ}\) to the horizontal. An observer notes only the vertical motion of the car. What is the car's (a) vertical velocity magnitude and (b) vertical travel distance?

A baseball player hits a home run into the right field upper deck. The ball lands in a row that is \(135 \mathrm{~m}\) horizontally from home plate and \(25.0 \mathrm{~m}\) above the playing field. An avid fan measures its time of flight to be \(4.10 \mathrm{~s}\). (a) Determine the ball's average velocity components. (b) Determine the magnitude and angle of its average velocity. (c) Explain why you cannot determine its average speed from the data given.

Using the triangle method, show graphically that (a) \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{A}}\) and \((\mathrm{b})\) if \(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{C}},\) then \(\overrightarrow{\mathbf{A}}=\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}\)

(a) Is vector addition associative? That is, does \((\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}})+\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+(\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}) ?(\mathbf{b})\) Justify your answer graphically.

The vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are perpendicular to one another and in the same plane. Prove that \(\overrightarrow{\mathbf{A}}_{x} \overrightarrow{\mathbf{B}}_{x}+\overrightarrow{\mathbf{A}}_{y} \overrightarrow{\mathbf{B}}_{y}=0\).

(a) What is the sum of \(\overrightarrow{\mathbf{A}}=3.0 \hat{\mathbf{x}}+5.0 \hat{\mathbf{y}}\) and \(\overrightarrow{\mathbf{B}}=1.0 \hat{\mathbf{x}}-3.0 \hat{\mathbf{y}}\) ? (b) What are the magnitude and direction of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) ?

For the two vectors \(\overrightarrow{\mathbf{x}}_{1}=(20 \mathrm{~m}) \hat{\mathbf{x}}\) and \(\overrightarrow{\mathbf{x}}_{2}=(15 \mathrm{~m}) \hat{\mathbf{x}}\) compute and show graphically (a) \(\overrightarrow{\mathbf{x}}_{1}+\overrightarrow{\mathbf{x}}_{2},\) (b) \(\overrightarrow{\mathbf{x}}_{1}-\overrightarrow{\mathbf{x}}_{2}\), and \((\mathrm{c}) \overrightarrow{\mathrm{x}}_{2}-\overrightarrow{\mathrm{x}}_{1}\)

For each of the given vectors, give a vector that, when added to it, yields a null vector (a vector with a magnitude of zero). Express the vector in the form other than that in which it is given (component or magnitudeangle): (a) \(\overrightarrow{\mathrm{A}}=4.5 \mathrm{~cm}, 40^{\circ}\) above the \(+x\) -axis; (b) \(\overrightarrow{\mathbf{B}}=(2.0 \mathrm{~cm}) \hat{\mathbf{x}}-(4.0 \mathrm{~cm}) \hat{\mathbf{y}} ;(\mathrm{c}) \overrightarrow{\mathrm{C}}=8.0 \mathrm{~cm}\) at an angle of \(60^{\circ}\) above the \(-x\) -axis.

(a) If each of the two components \((x\) and \(y\) ) of a vector are doubled, (1) the vector's magnitude doubles, but the direction remains unchanged; (2) the vector's magnitude remains unchanged, but the direction angle doubles; or (3) both the vector's magnitude and direction angle double. (b) If the \(x\) - and \(y\) -components of a vector of \(10 \mathrm{~m}\) at \(45^{\circ}\) are tripled, what is the new vector?

Two vectors are given by \(\overrightarrow{\mathbf{A}}=4.0 \hat{\mathbf{x}}-2.0 \hat{\mathbf{y}}\) and \(\overrightarrow{\mathbf{B}}=1.0 \hat{\mathbf{x}}+5.0 \hat{\mathbf{y}} .\) What is \((\) a) \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}},(\mathbf{b}) \overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}},\) and (c) a vector \(\overrightarrow{\mathbf{C}}\) such that \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=0\) ?

Given two vectors, \(\overrightarrow{\mathrm{A}}\) which has a length of 10.0 and makes an angle of \(45^{\circ}\) below the \(-x\) -axis, and \(\overrightarrow{\mathbf{B}}\) which has an \(x\) -component of +2.0 and \(\mathrm{a} y\) -component of +4.0 (a) sketch the vectors on \(x-y\) axes, with all their "tails" starting at the origin, and (b) calculate \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\).

The velocity of object 1 in component form is \(\overrightarrow{\mathbf{v}}_{1}=(+2.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{x}}+(-4.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{y}} .\) Object 2 has twice the speed of object 1 but moves in the opposite direction. (a) Determine the velocity of object 2 in component notation. (b) What is the speed of object \(2 ?\)

Given two vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) with magnitudes \(A\) and \(B\), respectively, you can subtract \(\overrightarrow{\mathbf{B}}\) from \(\overrightarrow{\mathbf{A}}\) to get a third vector \(\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\). If the magnitude of \(\overrightarrow{\mathbf{C}}\) is equal to \(C=A+B\), what is the relative orientation of vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathbf{B}}\) ?

In two successive chess moves, a player first moves his queen two squares forward, then moves the queen three steps to the left (from the player's view). Assume each square is \(3.0 \mathrm{~cm}\) on a side. (a) Using forward (toward the player's opponent) as the positive \(y\) -axis and right as the positive \(x\) -axis, write the queen's net displacement in component form. (b) At what net angle was the queen moved relative to the leftward direction?

Two force vectors, \(\overrightarrow{\mathbf{F}}_{1}=(3.0 \mathrm{~N}) \hat{\mathbf{x}}-(4.0 \mathrm{~N}) \hat{\mathbf{y}}\) and \(\overrightarrow{\mathbf{F}}_{2}=(-6.0 \mathrm{~N}) \hat{\mathbf{x}}+(4.5 \mathrm{~N}) \hat{\mathbf{y}},\) are applied to a particle. What third force \(\overrightarrow{\mathbf{F}}_{3}\) would make the net, or resultant, force on the particle zero?

A student works three problems involving the addition of two different vectors \(\overrightarrow{\mathbf{F}}_{1}\) and \(\overrightarrow{\mathbf{F}}_{2} .\) He states that the magnitudes of the three resultants are given by (a) \(F_{1}+F_{2}\), (b) \(F_{1}-F_{2},\) and (c) \(\sqrt{F_{1}^{2}+F_{2}^{2}}\). Are these results possible? If so, describe the vectors in each case.

Two displacements, one with a magnitude of \(15.0 \mathrm{~m}\) and a second with a magnitude of \(20.0 \mathrm{~m},\) can have any angle you want. (a) How would you create the sum of these two vectors so it has the largest magnitude possible? What is that magnitude? (b) How would you orient them so the magnitude of the sum was at its minimum? What value would that be? (c) Generalize the result to any two vectors.

A meteorologist tracks the movement of a thunderstorm with Doppler radar. At \(8: 00 \mathrm{PM},\) the storm was 60 mi northeast of her station. At 10: 00 PM, the storm is at 75 mi north. (a) The general direction of the thunderstorm's velocity is (1) south of east, (2) north of west, (3) north of east, (4) south of west. (b) What is the average velocity of the storm?

A flight controller determines that an airplane is 20.0 mi south of him. Half an hour later, the same plane is 35.0 mi northwest of him. (a) The general direction of the airplane's velocity is (1) east of south, (2) north of west, (3) north of east, (4) west of south. (b) If the plane is flying with constant velocity, what is its velocity during this time?

A golfer lines up for her first putt at a hole that is \(10.5 \mathrm{~m}\) exactly northwest of her ball's location. She hits the ball \(10.5 \mathrm{~m}\) and straight, but at the wrong angle, \(40^{\circ}\) from due north. In order for the golfer to have a "twoputt green," determine (a) the angle of the second putt and (b) the magnitude of the second putt's displacement. (c) Determine why you cannot determine the length of travel of the second putt.

A ball with a horizontal speed of \(1.0 \mathrm{~m} / \mathrm{s}\) rolls off a bench \(2.0 \mathrm{~m}\) high. (a) How long will the ball take to reach the floor? (b) How far from a point on the floor directly below the edge of the bench will the ball land?

An electron is ejected horizontally at a speed of \(1.5 \times 10^{6} \mathrm{~m} / \mathrm{s}\) from the electron gun of a computer monitor. If the viewing screen is \(35 \mathrm{~cm}\) from the end of the gun, how far will the electron travel in the vertical direction before hitting the screen? Based on your answer, do you think designers need to worry about this gravitational effect?

A ball rolls horizontally with a speed of \(7.6 \mathrm{~m} / \mathrm{s}\) off the edge of a tall platform. If the ball lands \(8.7 \mathrm{~m}\) from the point on the ground directly below the edge of the platform, what is the height of the platform?

A ball is projected horizontally with an initial speed of \(5.0 \mathrm{~m} / \mathrm{s}\). Find its (a) position and (b) velocity at \(t=2.5 \mathrm{~s}\)

An artillery crew wants to shell a position on level ground \(35 \mathrm{~km}\) away. If the gun has a muzzle velocity of \(770 \mathrm{~m} / \mathrm{s}\), to what angle of elevation should the gun be raised?

A pitcher throws a fastball horizontally at a speed of \(140 \mathrm{~km} / \mathrm{h}\) toward home plate, \(18.4 \mathrm{~m}\) away. \((\mathrm{a})\) If the batter's combined reaction and swing times total \(0.350 \mathrm{~s}\), how long can the batter watch the ball after it has left the pitcher's hand before swinging? (b) In traveling to the plate, how far does the ball drop from its original horizontal line?

Ball A rolls at a constant speed of \(0.25 \mathrm{~m} / \mathrm{s}\) on a table \(0.95 \mathrm{~m}\) above the floor, and ball \(\mathrm{B}\) rolls on the floor directly under the first ball with the same speed and direction. (a) When ball A rolls off the table and hits the floor, (1) ball B is ahead of ball A, (2) ball B collides with ball \(A,(3)\) ball \(A\) is ahead of ball \(B\). Why? (b) When ball A hits the floor, how far from the point directly below the edge of the table will each ball be?

A good-guy stuntman is being chased by bad guys on a building's level roof. He comes to the edge and is to jump to the level roof of a lower building \(4.0 \mathrm{~m}\) below and \(5.0 \mathrm{~m}\) away. What is the minimum launch speed the stuntman needs to complete the jump? (Landing on the edge is assumed complete.)

An astronaut on the Moon fires a projectile from a launcher on a level surface so as to get the maximum range. If the launcher gives the projectile a muzzle velocity of \(25 \mathrm{~m} / \mathrm{s},\) what is the range of the projectile? [Hint: The acceleration due to gravity on the Moon is only onesixth of that on the Earth.]

In 2004 two Martian probes successfully landed on the Red Planet. The final phase of the landing involved bouncing the probes until they came to rest (they were surrounded by protective inflated "balloons"). During one of the bounces, the telemetry (electronic data sent back to Earth) indicated that the probe took off at \(25.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(20^{\circ}\) and landed \(110 \mathrm{~m}\) away \((\) and then bounced again \()\) Assuming the landing region was level, determine the acceleration due to gravity near the Martian surface.

In laboratory situations, a projectile's range can be used to determine its speed. To see how this is done, suppose a ball rolls off a horizontal table and lands \(1.5 \mathrm{~m}\) out from the edge of the table. If the tabletop is \(90 \mathrm{~cm}\) above the floor, determine (a) the time the ball is in the air, and (b) the ball's speed as it left the table top.

A stone thrown off a bridge \(20 \mathrm{~m}\) above a river has an initial velocity of \(12 \mathrm{~m} / \mathrm{s}\) at an angle of \(45^{\circ}\) above the horizontal (vFig. 3.33). (a) What is the range of the stone? (b) At what velocity does the stone strike the water?

If the maximum height reached by a projectile launched on level ground is equal to half the projectile's range, what is the launch angle?

A shot-putter launches the shot from a vertical distance of \(2.0 \mathrm{~m}\) off the ground (from just above her ear) at a speed of \(12.0 \mathrm{~m} / \mathrm{s} .\) The initial velocity is at an angle of \(20^{\circ}\) above the horizontal. Assume the ground is flat. (a) Compared to a projectile launched at the same angle and speed at ground level, would the shot be in the air (1) a longer time, (2) a shorter time, or (3) the same amount of time? (b) Justify your answer explicitly; determine the shot's range and velocity just before impact in unit vector (component) notation.

A quarterback passes a football-at a velocity of \(50 \mathrm{ft} / \mathrm{s}\) at an angle of \(40^{\circ}\) to the horizontal-toward an intended receiver 30 yd downfield. The pass is released \(5.0 \mathrm{ft}\) above the ground. Assume that the receiver is stationary and that he will catch the ball if it comes to him. Will the pass be completed? If not, will the throw be long or short?

A 2.05 -m-tall basketball player takes a shot when he is \(6.02 \mathrm{~m}\) from the basket (at the three-point line). If the launch angle is \(25^{\circ}\) and the ball was launched at the level of the player's head, what must be the release speed of the ball for the player to make the shot? The basket is \(3.05 \mathrm{~m}\) above the floor.

While you are traveling in a car on a straight, level interstate highway at \(90 \mathrm{~km} / \mathrm{h}\), another car passes you in the same direction; its speedometer reads \(120 \mathrm{~km} / \mathrm{h}\). (a) What is your velocity relative to the other driver? (b) What is the other car's velocity relative to you?