Chapter 4

The position vector for an electron is \(\vec{r}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}-\) \((3.0 \mathrm{~m}) \hat{\mathrm{j}}+(2.0 \mathrm{~m}) \hat{\mathrm{k}}\). (a) Find the magnitude of \(\vec{r}\). (b) Sketch the vector on a right-handed coordinate system.

-1 The position vector for an electron is \(\vec{r}=(5.0 \mathrm{~m}) \hat{\mathrm{i}}-\) \((3.0 \mathrm{~m}) \hat{\mathrm{j}}+(2.0 \mathrm{~m}) \hat{\mathrm{k}}\). (a) Find the magnitude of \(\vec{r}\). (b) Sketch the vector on a right-handed coordinate system. -2 A watermelon seed has the following coordinates: \(x=-5.0 \mathrm{~m}\), \(y=8.0 \mathrm{~m}\), and \(z=0 \mathrm{~m} .\) Find its position vector (a) in unit-vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the \(x\) axis. (d) Sketch the vector on a right-handed coordinate system. If the seed is moved to the \(x y z\) coordinates \((3.00 \mathrm{~m},\), \(0 \mathrm{~m}, 0 \mathrm{~m}\) ), what is its displacement (e) in unit-vector notation and as (f) a magnitude and (g) an angle relative to the positive \(x\) direction?

A positron undergoes a displacement \(\Delta \vec{r}=2.0 \hat{\mathrm{i}}-3.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}}\), ending with the position vector \(\vec{r}=3.0 \hat{\mathrm{j}}-4.0 \hat{\mathrm{k}}\), in meters. What was the positron's initial position vector? w.4 The minute hand of a wall clock measures \(10 \mathrm{~cm}\) from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

The minute hand of a wall clock measures \(10 \mathrm{~cm}\) from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

A train at a constant \(60.0 \mathrm{~km} / \mathrm{h}\) moves east for \(40.0 \mathrm{~min}\), then in a direction \(50.0^{\circ}\) east of due north for \(20.0 \mathrm{~min}\), and then west for \(50.0 \mathrm{~min}\). What are the (a) magnitude and (b) angle of its average velocity during this trip?

An electron's position is given by \(\vec{r}=3.00 t \hat{\mathrm{i}}-4.00 t \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}}\), with \(t\) in seconds and \(\vec{r}\) in meters. (a) In unit-vector notation, what js the electron's velocity \(\vec{v}(t) ?\) At \(t=2.00 \mathrm{~s}\), what is \(\vec{v}(\mathrm{~b})\) in unitvector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the \(x\) axis?

An ion's position vector is initially \(\vec{r}=5.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\), and \(10 \mathrm{~s}\) later it is \(\vec{r}=-2.0 \hat{\mathrm{i}}+8.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}\), all in meters. In unit- vector notation, what is its \(\vec{v}_{\text {avg }}\) during the \(10 \mathrm{~s}\) ?

A plane flies \(483 \mathrm{~km}\) east from city \(A\) to city \(B\) in \(45.0 \mathrm{~min}\) and then \(966 \mathrm{~km}\) south from city \(B\) to city \(C\) in \(1.50 \mathrm{~h}\). For the total trip, what are the (a) magnitude and (b) direction of the plane's displacement, the (c) magnitude and (d) direction of its average velocity, and (e) its average speed?

The position \(\vec{r}\) of a particle moving in an \(x y\) plane is given by \(\vec{r}=\left(2.00 t^{3}-5.00 t\right) \hat{\mathrm{i}}+\left(6.00-7.00 t^{4}\right) \hat{\mathrm{j}}\), with \(\vec{r}\) in meters and \(\mathrm{t}\) in seconds. In unit-vector notation, calculate (a) \(\vec{r},(\mathrm{~b}) \vec{v}\), and \((\mathrm{c}) \vec{a}\) for \(t=2.00 \mathrm{~s}\) (d) What is the angle between the positive direction of the \(x\) axis and a line tangent to the particle's path at \(t=2.00 \mathrm{~s}\) ?

At one instant a bicyclist is \(40.0 \mathrm{~m}\) due east of a park's flagpole, going due south with a speed of \(10.0 \mathrm{~m} / \mathrm{s}\). Then \(30.0 \mathrm{~s}\) later, the cyclist is \(40.0 \mathrm{~m}\) due north of the flagpole, going due east with a speed of \(10.0 \mathrm{~m} / \mathrm{s}\). For the cyclist in this \(30.0\) s interval, what are the (a) magnitude and (b) direction of the displacement, the (c) magnitude and (d) direction of the average velocity, and the (e) magnitude and (f) direction of the average acceleration?

A particle moves so that its position (in meters) as a function of time (in seconds) is \(\vec{r}=\hat{\mathrm{i}}+4 t^{2 \hat{\mathrm{j}}}+t \hat{\mathrm{k}}\). Write expressions for (a) its velocity and (b) its acceleration as functions of time.

A proton initially has \(\vec{v}=4.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}\) and then \(4.0 \mathrm{~s}\) later has \(\vec{v}=-2.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+5.0 \hat{\mathrm{k}}\) (in meters per second). For that \(4.0 \mathrm{~s}\), what are (a) the proton's average acceleration \(\vec{a}_{\text {avg }}\) in unitvector notation, (b) the magnitude of \(\vec{a}_{\text {avg }}\), and \((\) c) the angle between \(\vec{a}_{\text {avg }}\) and the positive direction of the \(x\) axis?

A particle leaves the origin with an initial velocity \(\vec{v}=(3.00 \hat{\mathrm{i}}) \mathrm{m} / \mathrm{s}\) and a constant acceleration \(\vec{a}=(-1.00 \hat{\mathrm{i}}-\) \(0.500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}\). When it reaches its maximum \(x\) coordinate, what are its (a) velocity and (b) position vector?

The velocity \(\vec{v}\) of a particle moving in the \(x y\) plane is given by \(\vec{v}=\left(6.0 t-4.0 t^{2}\right) \hat{\mathrm{i}}+8.0 \hat{\mathrm{j}}\), with \(\vec{v}\) in meters per second and \(t(>0)\) in seconds. (a) What is the acceleration when \(t=3.0 \mathrm{~s}\) ? (b) When (if ever) is the acceleration zero? (c) When (if ever) is the velocity zero? (d) When (if ever) does the speed equal \(10 \mathrm{~m} / \mathrm{s} ?\)

A cart is propelled over an \(x y\) plane with acceleration components \(a_{x}=4.0 \mathrm{~m} / \mathrm{s}^{2}\) and \(a_{y}=-2.0 \mathrm{~m} / \mathrm{s}^{2} .\) Its initial velocity has components \(v_{0 x}=8.0 \mathrm{~m} / \mathrm{s}\) and \(v_{0 y}=12 \mathrm{~m} / \mathrm{s}\). In unit-vector notation, what is the velocity of the cart when it reaches its greatest \(y\) coordinate? ..18 A moderate wind accelerates a pebble over a horizontal \(x y\) plane with a constant acceleration \(\vec{a}=\left(5.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(7.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\).

A moderate wind accelerates a pebble over a horizontal \(x y\) plane with a constant acceleration \(\vec{a}=\left(5.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(7.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). At time \(t=0\), the velocity is \((4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}\). What are the (a) magnitude and (b) angle of its velocity when it has been displaced by \(12.0 \mathrm{~m}\) parallel to the \(x\) axis?

The acceleration of a particle moving only on a horizontal \(x y\) plane is given by \(\vec{a}=3 t \hat{i}+4 t \hat{j}\), where \(\vec{a}\) is in meters per secondsquared and \(t\) is in seconds. At \(t=0\), the position vector \(\vec{r}=(20.0 \mathrm{~m}) \hat{\mathrm{i}}+(40.0 \mathrm{~m}) \hat{\mathrm{j}}\) locates the particle, which then has the velocity vector \(\vec{v}=(5.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(2.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\). At \(t=4.00 \mathrm{~s}\), what are (a) its position vector in unit-vector notation and (b) the angle between its direction of travel and the positive direction of the \(x\) axis?

A dart is thrown horizontally with an initial speed of \(10 \mathrm{~m} / \mathrm{s}\) toward point \(P\), the bull's-eye on a dart board. It hits at point \(Q\) on the rim, vertically below \(P, 0.19 \mathrm{~s}\) later. (a) What is the distance \(P Q ?\) (b) How far away from the dart board is the dart released?

A small ball rolls horizontally off the edge of a tabletop that is \(1.20 \mathrm{~m}\) high. It strikes the floor at a point \(1.52 \mathrm{~m}\) horizontally from the table edge. (a) How long is the ball in the air? (b) What is its speed at the instant it leaves the table?

A projectile is fired horizontally from a gun that is \(45.0 \mathrm{~m}\) above flat ground, emerging from the gun with a speed of \(250 \mathrm{~m} / \mathrm{s}\). (a) How long does the projectile remain in the air? (b) At what horizontal distance from the firing point does it strike the ground? (c) What is the magnitude of the vertical component of its velocity as it strikes the ground?

The current world-record motorcycle jump is \(77.0 \mathrm{~m}\), set by Jason Renie. Assume that he left the take-off ramp at \(12.0^{\circ}\) to the horizontal and that the take-off and landing heights are the same. Neglecting air drag, determine his take-off speed.

A projectile's launch speed is five times its speed at maximum height. Find launch angle \(\theta_{0}\).

A soccer ball is kicked from the ground with an initial speed of \(19.5 \mathrm{~m} / \mathrm{s}\) at an upward angle of \(45^{\circ} .\) A player \(55 \mathrm{~m}\) away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground?

In a jump spike, a volleyball player slams the ball from overhead and toward the opposite floor. Controlling the angle of the spike is difficult. Suppose a ball is spiked from a height of \(2.30\) \(\mathrm{m}\) with an initial speed of \(20.0 \mathrm{~m} / \mathrm{s}\) at a downward angle of \(18.00^{\circ} .\) How much farther on the opposite floor would it have landed if the downward angle were, instead, \(8.00^{\circ} ?\)

You throw a ball toward a wall at speed \(25.0 \mathrm{~m} / \mathrm{s}\) and at angle \(\theta_{0}=40.0^{\circ}\) above the horizontal (Fig. 4-35). The wall is distance \(d=\) \(22.0 \mathrm{~m}\) from the release point of the ball. (a) How far above the release point does the ball hit the wall? What are the (b) horizontal and (c) vertical components of its velocity as it hits the wall? (d) When it hits, has it passed the highest point on its trajectory?

A plane, diving with constant speed at an angle of \(53.0^{\circ}\) with the vertical, releases a projectile at an altitude of \(730 \mathrm{~m}\). The projectile hits the ground \(5.00 \mathrm{~s}\) after release. (a) What is the speed of the plane? (b) How far does the projectile travel horizontally during its flight? What are the (c) horizontal and (d) vertical components of its velocity just before striking the ground?

A rifle that shoots bullets at \(460 \mathrm{~m} / \mathrm{s}\) is to be aimed at a target \(45.7 \mathrm{~m}\) away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?

During a tennis match, a player serves the ball at \(23.6 \mathrm{~m} / \mathrm{s}\), with the center of the ball leaving the racquet horizontally \(2.37 \mathrm{~m}\) above the court surface. The net is \(12 \mathrm{~m}\) away and \(0.90 \mathrm{~m}\) high. When the ball reaches the net, (a) does the ball clear it and (b) what is the distance between the center of the ball and the top of the net? Suppose that, instead, the ball is served as before but now it leaves the racquet at \(5.00^{\circ}\) below the horizontal. When the ball reaches the net, (c) does the ball clear it and (d) what now is the distance between the center of the ball and the top of the net?

A lowly high diver pushes off horizontally with a speed of \(2.00 \mathrm{~m} / \mathrm{s}\) from the platform edge \(10.0 \mathrm{~m}\) above the surface of the water. (a) At what horizontal distance from the edge is the diver \(0.800 \mathrm{~s}\) after pushing off? (b) At what vertical distance above the surface of the water is the diver just then? (c) At what horizontal distance from the edge does the diver strike the water?

Suppose that a shot putter can put a shot at the worldclass speed \(v_{0}=15.00 \mathrm{~m} / \mathrm{s}\) and at a height of \(2.160 \mathrm{~m}\). What horizontal distance would the shot travel if the launch angle \(\theta_{0}\) is (a) \(45.00^{\circ}\) and (b) \(42.00^{\circ}\) ? The answers indicate that the angle of \(45^{\circ}\), which maximizes the range of projectile motion, does not maximize the horizontal distance when the launch and landing are at different heights.

A baseball leaves a pitcher's hand horizontally at a speed of \(161 \mathrm{~km} / \mathrm{h}\). The distance to the batter is \(18.3 \mathrm{~m}\). (a) How long does the ball take to travel the first half of that distance? (b) The second half? (c) How far does the ball fall freely during the first half? (d) During the second half? (e) Why aren't the quantities in (c) and (d) equal?

In basketball, hang is an illusion in which a player seems to weaken the gravitational acceleration while in midair. The illusion depends much on a skilled player's ability to rapidly shift the ball between hands during the flight, but it might also be supported by the longer horizontal distance the player travels in the upper part of the jump than in the lower part. If a player jumps with an initial speed of \(v_{0}=7.00 \mathrm{~m} / \mathrm{s}\) at an angle of \(\theta_{0}=35.0^{\circ}\), what percent of the jump's range does the player spend in the upper half of the jump (between maximum height and half maximum height \() ?\)

A batter hits a pitched ball when the center of the ball is \(1.22 \mathrm{~m}\) above the ground. The ball leaves the bat at an angle of \(45^{\circ}\) with the ground. With that launch, the ball should have a horizontal range (returning to the launch level) of \(107 \mathrm{~m}\). (a) Does the ball clear a \(7.32\) -m-high fence that is \(97.5 \mathrm{~m}\) horizontally from the launch point? (b) At the fence, what is the distance between the fence top and the ball center?

A football kicker can give the ball an initial speed of \(25 \mathrm{~m} / \mathrm{s}\). What are the (a) least and (b) greatest elevation angles at which he can kick the ball to score a field goal from a point \(50 \mathrm{~m}\) in front of goalposts whose horizontal bar is \(3.44 \mathrm{~m}\) above the ground?

Two seconds after being projected from ground level, a projectile is displaced \(40 \mathrm{~m}\) horizontally and \(53 \mathrm{~m}\) vertically above its launch point. What are the (a) horizontal and (b) vertical components of the initial velocity of the projectile? (c) At the instant the projectile achieves its maximum height above ground level, how far is it displaced horizontally from the launch point?

A ball rolls horizontally off the top of a stairway with a speed of \(1.52 \mathrm{~m} / \mathrm{s}\). The steps are \(20.3 \mathrm{~cm}\) high and \(20.3 \mathrm{~cm}\) wide. Which step does the ball hit first?

An Earth satellite moves in a circular orbit \(640 \mathrm{~km}\) (uniform circular motion) above Earth's surface with a period of \(98.0 \mathrm{~min}\). What are (a) the speed and (b) the magnitude of the centripetal acceleration of the satellite?

A carnival merry-go-round rotates about a vertical axis at a constant rate. A man standing on the edge has a constant speed of \(3.66 \mathrm{~m} / \mathrm{s}\) and a centripetal acceleration \(\vec{a}\) of magnitude \(1.83 \mathrm{~m} / \mathrm{s}^{2}\) Position vector \(\vec{r}\) locates him relative to the rotation axis. (a) What is the magnitude of \(\vec{r}\) ? What is the direction of \(\vec{r}\) when \(\vec{a}\) is directed (b) due east and (c) due south?

A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of \(0.15 \mathrm{~m}\). (a) Through what distance does the tip move in one revolution? What are (b) the tip's speed and (c) the magnitude of its acceleration? (d) What is the period of the motion?

A woman rides a carnival Ferris wheel at radius \(15 \mathrm{~m}\), completing five turns about its horizontal axis every minute. What are (a) the period of the motion, the (b) magnitude and (c) direction of her centripetal acceleration at the highest point, and the (d) magnitude and (e) direction of her centripetal acceleration at the lowest point?

A centripetal-acceleration addict rides in uniform circular motion with radius \(r=3.00 \mathrm{~m}\). At one instant his acceleration is \(\vec{a}=\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). At that instant, what are the val- ues of (a) \(\vec{v} \cdot \vec{a}\) and (b) \(\vec{r} \times \vec{a}\) ?

When a large star becomes a supernova, its core may be compressed so tightly that it becomes a neutron star, with a radius of about \(20 \mathrm{~km}\) (about the size of the San Francisco area). If a neutron star rotates once every second, (a) what is the speed of a particle on the star's equator and (b) what is the magnitude of the particle's centripetal acceleration? (c) If the neutron star rotates faster, do the answers to (a) and (b) increase, decrease, or remain the same?

What is the magnitude of the acceleration of a sprinter running at \(10 \mathrm{~m} / \mathrm{s}\) when rounding a turn of radius \(25 \mathrm{~m}\) ?

At \(t_{1}=2.00 \mathrm{~s}\), the acceleration of a particle in counterclockwise circular motion is \(\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). It moves at constant speed. At time \(t_{2}=5.00 \mathrm{~s}\), the particle's acceleration is \(\left(4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). What is the radius of the path taken by the particle if \(t_{2}-t_{1}\) is less than one period?

A particle moves horizontally in uniform circular motion, over a horizontal \(x y\) plane. At one instant, it moves through the point at coordinates \((4.00 \mathrm{~m}, 4.00 \mathrm{~m})\) with a velocity of \(-5.00 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}\) and an acceleration of \(+12.5 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2} .\) What are the (a) \(x\) and (b) \(y\) coordinates of the center of the circular path?

A particle moves along a circular path over a horizontal \(x y\) coordinate system, at constant speed. At time \(t_{1}=4.00 \mathrm{~s}\), it is at point \((5.00 \mathrm{~m}, 6.00 \mathrm{~m})\) with velocity \((3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\) and acceleration in the positive \(x\) direction. At time \(t_{2}=10.0 \mathrm{~s}\), it has velocity \((-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}\) and acceleration in the positive \(y\) direction. What are the (a) \(x\) and (b) \(y\) coordinates of the center of the circular path if \(t_{2}-t_{1}\) is less than one period?

A boy whirls a stone in a horizontal circle of radius \(1.5 \mathrm{~m}\) and at height \(2.0 \mathrm{~m}\) above level ground. The string breaks, and the stone flies off horizontally and strikes the ground after traveling a horizontal distance of \(10 \mathrm{~m}\). What is the magnitude of the centripetal acceleration of the stone during the circular motion?

A cat rides a merry-go-round turning with uniform circular motion. At time \(t_{1}=2.00 \mathrm{~s}\), the cat's velocity is \(\vec{v}_{1}=\) \((3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\), measured on a horizontal \(x y\) coordinate system. At \(t_{2}=5.00 \mathrm{~s}\), the cat's velocity is \(\vec{v}_{2}=(-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+\) \((-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\). What are (a) the magnitude of the cat's centripetal acceleration and (b) the cat's average acceleration during the time interval \(t_{2}-t_{1}\), which is less than one period?

A cameraman on a pickup truck is traveling westward at \(20 \mathrm{~km} / \mathrm{h}\) while he records a cheetah that is moving westward \(30 \mathrm{~km} / \mathrm{h}\) faster than the truck. Suddenly, the cheetah stops, turns, and then runs at \(45 \mathrm{~km} / \mathrm{h}\) eastward, as measured by a suddenly nervous crew member who stands alongside the cheetah's path. The change in the animal's velocity takes \(2.0 \mathrm{~s}\). What are the (a) magnitude and (b) direction of the animal's acceleration according to the cameraman and the (c) magnitude and (d) direction according to the nervous crew member?

A boat is traveling upstream in the positive direction of an \(x\) axis at \(14 \mathrm{~km} / \mathrm{h}\) with respect to the water of a river. The water is flowing at \(9.0 \mathrm{~km} / \mathrm{h}\) with respect to the ground. What are the (a) magnitude and (b) direction of the boat's velocity with respect to the ground? A child on the boat walks from front to rear at \(6.0 \mathrm{~km} / \mathrm{h}\) with respect to the boat. What are the (c) magnitude and (d) direction of the child's velocity with respect to the ground?