### Problems

## 4.1 Displacement and Velocity Vectors

The coordinates of a particle in a rectangular coordinate system are (1.0, –4.0, 6.0). What is the position vector of the particle?

The position of a particle changes from ${\overrightarrow{r}}_{1}=(2.0\text{}\widehat{i}+3.0\widehat{j})\text{cm}$ to ${\overrightarrow{r}}_{2}=(\mathrm{-4.0}\widehat{i}+3.0\widehat{j})\phantom{\rule{0.2em}{0ex}}\text{cm}.$ What is the particle’s displacement?

The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 m. The fairway off the tee is taken to be the *x* direction. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement $\text{\Delta}{\overrightarrow{r}}_{1}=300.0\phantom{\rule{0.2em}{0ex}}\text{m}\widehat{i},$ and hits his second shot 189.0 m with a displacement $\text{\Delta}{\overrightarrow{r}}_{2}=172.0\phantom{\rule{0.2em}{0ex}}\text{m}\widehat{i}+80.3\phantom{\rule{0.2em}{0ex}}\text{m}\widehat{j}.$ What is the final displacement of the golf ball from where it started?

A bird flies straight northeast a distance of 95.0 km for 3.0 h. With the *x*-axis due east and the *y*-axis due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?

A cyclist rides 5.0 km due east, then 10.0 km $20\text{\xb0}$ west of north. From this point she rides 8.0 km due west. What is the final displacement from where the cyclist started?

New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and $45\text{\xb0}$ from straight down the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him 10 m away. What is the final displacement of the puck? See the following figure.

The position of a particle is $\overrightarrow{r}(t)=4.0{t}^{2}\widehat{i}-3.0\widehat{j}+2.0{t}^{3}\widehat{k}\text{m}.$ (a) What is the velocity of the particle at 0 s and at $1.0$ s? (b) What is the average velocity between 0 s and $1.0$ s?

Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at $45\text{\xb0}$ with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at $90\text{\xb0}$ with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?

The F-35B Lighting II is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to 20.00-m height above the ground and then follows a flight path angled at $30\text{\xb0}$ with respect to the ground for 20.00 km, what is the final displacement?

## 4.2 Acceleration Vector

The position of a particle is $\overrightarrow{r}(t)=(3.0{t}^{2}\widehat{i}+5.0\widehat{j}-6.0t\widehat{k})\phantom{\rule{0.2em}{0ex}}\text{m}.$ (a) Determine its velocity and acceleration as functions of time. (b) What are its velocity and acceleration at time *t* = 0?

A particle’s acceleration is $(4.0\widehat{i}+3.0\widehat{j})\text{m/}{\text{s}}^{2}.$ At *t* = 0, its position and velocity are zero. (a) What are the particle’s position and velocity as functions of time? (b) Find the equation of the path of the particle. Draw the *x-* and *y-*axes and sketch the trajectory of the particle.

A boat leaves the dock at *t* = 0 and heads out into a lake with an acceleration of $2.0\phantom{\rule{0.2em}{0ex}}\text{m/}{\text{s}}^{2}\widehat{i}.$ A strong wind is pushing the boat, giving it an additional velocity of $2.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\widehat{i}+1.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\widehat{j}.$ (a) What is the velocity of the boat at *t* = 10 s? (b) What is the position of the boat at *t* = 10s? Draw a sketch of the boat’s trajectory and position at *t* = 10 s, showing the *x-* and *y*-axes.

The position of a particle for *t* > 0 is given by $\overrightarrow{r}(t)=(3.0{t}^{2}\widehat{i}-7.0{t}^{3}\widehat{j}-5.0{t}^{\mathrm{-2}}\widehat{k})\phantom{\rule{0.2em}{0ex}}\text{m}.$ (a) What is the velocity as a function of time? (b) What is the acceleration as a function of time? (c) What is the particle’s velocity at *t* = 2.0 s? (d) What is its speed at *t* = 1.0 s and *t* = 3.0 s? (e) What is the average velocity between *t* = 1.0 s and *t* = 2.0 s?

The acceleration of a particle is a constant. At *t* = 0 the velocity of the particle is $(10\widehat{i}+20\widehat{j})\text{m/s}.$ At *t* = 4 s the velocity is $10\widehat{j}\text{m/s}.$ (a) What is the particle’s acceleration? (b) How do the position and velocity vary with time? Assume the particle is initially at the origin.

A particle has a position function $\overrightarrow{r}(t)=\text{cos}(1.0t)\widehat{i}+\text{sin}(1.0t)\widehat{j}+t\widehat{k},$ where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?

A Lockheed Martin F-35 II Lighting jet takes off from an aircraft carrier with a runway length of 90 m and a takeoff speed 70 m/s at the end of the runway. Jets are catapulted into airspace from the deck of an aircraft carrier with two sources of propulsion: the jet propulsion and the catapult. At the point of leaving the deck of the aircraft carrier, the F-35’s acceleration decreases to a constant acceleration of $5.0\phantom{\rule{0.2em}{0ex}}\text{m/}{\text{s}}^{2}$ at $30\text{\xb0}$ with respect to the horizontal. (a) What is the initial acceleration of the F-35 on the deck of the aircraft carrier to make it airborne? (b) Write the position and velocity of the F-35 in unit vector notation from the point it leaves the deck of the aircraft carrier. (c) At what altitude is the fighter 5.0 s after it leaves the deck of the aircraft carrier? (d) What is its velocity and speed at this time? (e) How far has it traveled horizontally?

## 4.3 Projectile Motion

A bullet is shot horizontally from shoulder height (1.5 m) with an initial speed 200 m/s. (a) How much time elapses before the bullet hits the ground? (b) How far does the bullet travel horizontally?

A marble rolls off a tabletop 1.0 m high and hits the floor at a point 3.0 m away from the table’s edge in the horizontal direction. (a) How long is the marble in the air? (b) What is the speed of the marble when it leaves the table’s edge? (c) What is its speed when it hits the floor?

A dart is thrown horizontally at a speed of 10 m/s at the bull’s-eye of a dartboard 2.4 m away, as in the following figure. (a) How far below the intended target does the dart hit? (b) What does your answer tell you about how proficient dart players throw their darts?

An airplane flying horizontally with a speed of 500 km/h at a height of 800 m drops a crate of supplies (see the following figure). If the parachute fails to open, how far in front of the release point does the crate hit the ground?

Suppose the airplane in the preceding problem fires a projectile horizontally in its direction of motion at a speed of 300 m/s relative to the plane. (a) How far in front of the release point does the projectile hit the ground? (b) What is its speed when it hits the ground?

A fastball pitcher can throw a baseball at a speed of 40 m/s (90 mi/h). (a) Assuming the pitcher can release the ball 16.7 m from home plate so the ball is moving horizontally, how long does it take the ball to reach home plate? (b) How far does the ball drop between the pitcher’s hand and home plate?

A projectile is launched at an angle of $30\text{\xb0}$ and lands 20 s later at the same height as it was launched. (a) What is the initial speed of the projectile? (b) What is the maximum altitude? (c) What is the range? (d) Calculate the displacement from the point of launch to the position on its trajectory at 15 s.

A basketball player shoots toward a basket 6.1 m away and 3.0 m above the floor. If the ball is released 1.8 m above the floor at an angle of $60\text{\xb0}$ above the horizontal, what must the initial speed be if it were to go through the basket?

At a particular instant, a hot air balloon is 100 m in the air and descending at a constant speed of 2.0 m/s. At this exact instant, a girl throws a ball horizontally, relative to herself, with an initial speed of 20 m/s. When she lands, where will she find the ball? Ignore air resistance.

A man on a motorcycle traveling at a uniform speed of 10 m/s throws an empty can straight upward relative to himself with an initial speed of 3.0 m/s. Find the equation of the trajectory as seen by a police officer on the side of the road. Assume the initial position of the can is the point where it is thrown. Ignore air resistance.

An athlete can jump a distance of 8.0 m in the broad jump. What is the maximum distance the athlete can jump on the Moon, where the gravitational acceleration is one-sixth that of Earth?

The maximum horizontal distance a boy can throw a ball is 50 m. Assume he can throw with the same initial speed at all angles. How high does he throw the ball when he throws it straight upward?

A rock is thrown off a cliff at an angle of $53\text{\xb0}$ with respect to the horizontal. The cliff is 100 m high. The initial speed of the rock is 30 m/s. (a) How high above the edge of the cliff does the rock rise? (b) How far has it moved horizontally when it is at maximum altitude? (c) How long after the release does it hit the ground? (d) What is the range of the rock? (e) What are the horizontal and vertical positions of the rock relative to the edge of the cliff at *t* = 2.0 s, *t* = 4.0 s, and *t* = 6.0 s?

Trying to escape his pursuers, a secret agent skis off a slope inclined at $30\text{\xb0}$ below the horizontal at 60 km/h. To survive and land on the snow 100 m below, he must clear a gorge 60 m wide. Does he make it? Ignore air resistance.

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of $40\text{\xb0}$ with an initial speed of 20 m/s, how close to the green does she come?

A projectile is shot at a hill, the base of which is 300 m away. The projectile is shot at $60\text{\xb0}$ above the horizontal with an initial speed of 75 m/s. The hill can be approximated by a plane sloped at $20\text{\xb0}$ to the horizontal. Relative to the coordinate system shown in the following figure, the equation of this straight line is $y=(\text{tan}20\text{\xb0})x-109.$ Where on the hill does the projectile land?

An astronaut on Mars kicks a soccer ball at an angle of $45\text{\xb0}$ with an initial velocity of 15 m/s. If the acceleration of gravity on Mars is $3.7\text{m/}{\text{s}}^{2}$, (a) what is the range of the soccer kick on a flat surface? (b) What would be the range of the same kick on the Moon, where gravity is one-sixth that of Earth?

Mike Powell holds the record for the long jump of 8.95 m, established in 1991. If he left the ground at an angle of $15\text{\xb0},$ what was his initial speed?

MIT’s robot cheetah can jump over obstacles 46 cm high and has speed of 12.0 km/h. (a) If the robot launches itself at an angle of $60\text{\xb0}$ at this speed, what is its maximum height? (b) What would the launch angle have to be to reach a height of 46 cm?

Mt. Asama, Japan, is an active volcano. In 2009, an eruption threw solid volcanic rocks that landed 1 km horizontally from the crater. If the volcanic rocks were launched at an angle of $40\text{\xb0}$ with respect to the horizontal and landed 900 m below the crater, (a) what would be their initial velocity and (b) what is their time of flight?

Drew Brees of the New Orleans Saints can throw a football 23.0 m/s (50 mph). If he angles the throw at $10\text{\xb0}$ from the horizontal, what distance does it go if it is to be caught at the same elevation as it was thrown?

The Lunar Roving Vehicle used in NASA’s late *Apollo* missions reached an unofficial lunar land speed of $5.0\phantom{\rule{0.2em}{0ex}}\text{m/s}$ by astronaut Eugene Cernan. If the rover was moving at this speed on a flat lunar surface and hit a small bump that projected it off the surface at an angle of $20\text{\xb0},$ how long would it be “airborne” on the Moon?

A soccer goal is 2.44 m high. A player kicks the ball at a distance 10 m from the goal at an angle of $25\text{\xb0}.$ The ball hits the crossbar at the top of the goal. What is the initial speed of the soccer ball?

Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of $3.7\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}.$

In 1999, Robbie Knievel was the first to jump the Grand Canyon on a motorcycle. At a narrow part of the canyon (69.0 m wide) and traveling 35.8 m/s off the takeoff ramp, he reached the other side. What was his launch angle?

You throw a baseball at an initial speed of 15.0 m/s at an angle of $30\text{\xb0}$ with respect to the horizontal. What would the ball’s initial speed have to be at $30\text{\xb0}$ on a planet that has twice the acceleration of gravity as Earth to achieve the same range? Consider launch and impact on a horizontal surface.

Aaron Rodgers throws a football at 20.0 m/s to his wide receiver, who is running straight down the field at 9.4 m/s. If Aaron throws the football when the wide receiver is 10.0 m in front of him, (a) at what angle does Aaron have to launch the ball so the ball will be at the same height as the receiver when the receiver makes it to 20.0 m in front of Aaron? (b) Will the receiver be able to catch the ball?

## 4.4 Uniform Circular Motion

A flywheel is rotating at 30 rev/s. What is the total angle, in radians, through which a point on the flywheel rotates in 40 s?

A particle travels in a circle of radius 10 m at a constant speed of 20 m/s. What is the magnitude of the acceleration?

Cam Newton of the Carolina Panthers throws a perfect football spiral at 8.0 rev/s. The radius of a pro football is 8.5 cm at the middle of the short side. What is the centripetal acceleration of the laces on the football?

A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00-m radius, at how many revolutions per minute are the riders subjected to a centripetal acceleration equal to that of gravity?

A runner taking part in the 200-m dash must run around the end of a track that has a circular arc with a radius of curvature of 30.0 m. The runner starts the race at a constant speed. If she completes the 200-m dash in 23.2 s and runs at constant speed throughout the race, what is her centripetal acceleration as she runs the curved portion of the track?

An experimental jet rocket travels around Earth along its equator just above its surface. At what speed must the jet travel if the magnitude of its acceleration is *g*?

A fan is rotating at a constant 360.0 rev/min. What is the magnitude of the acceleration of a point on one of its blades 10.0 cm from the axis of rotation?

A point located on the second hand of a large clock has a radial acceleration of $0.1\text{cm}\text{/}{\text{s}}^{2}.$ How far is the point from the axis of rotation of the second hand?

## 4.5 Relative Motion in One and Two Dimensions

The coordinate axes of the reference frame ${S}^{\prime}$ remain parallel to those of $S$, as ${S}^{\prime}$ moves away from $S$ at a constant velocity
${\overrightarrow{v}}_{{S}^{\prime}}^{S}=(4.0\widehat{i}+3.0\widehat{j}+5.0\widehat{k})\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}.$
(a) If at time *t* = 0 the origins coincide, what is the position of the origin ${O}^{\prime}$ in the *S* frame as a function of time? (b) How is particle position for $\overrightarrow{r}(t)$ and ${\overrightarrow{r}}^{\prime}(t),$ as measured in *S* and ${S}^{\prime},$ respectively, related? (c) What is the relationship between particle velocities $\overrightarrow{v}(t)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\overrightarrow{v}}^{\prime}(t)?$ (d) How are accelerations $\overrightarrow{a}\text{(t)}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\overrightarrow{a}}^{\prime}\text{(t)}$ related?

The coordinate axes of the reference frame ${S}^{\prime}$ remain parallel to those of *S*, as ${S}^{\prime}$ moves away from *S* at a constant velocity ${\overrightarrow{v}}_{{S}^{\prime}S}=(1.0\widehat{i}+2.0\widehat{j}+3.0\widehat{k})t\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}$. (a) If at time *t* = 0 the origins coincide, what is the position of origin ${O}^{\prime}$ in the *S* frame as a function of time? (b) How is particle position for $\overrightarrow{r}(t)$ and ${\overrightarrow{r}}^{\prime}(t)$, as measured in *S* and ${S}^{\prime},$ respectively, related? (c) What is the relationship between particle velocities $\overrightarrow{v}(t)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\overrightarrow{v}}^{\prime}(t)?$ (d) How are accelerations $\overrightarrow{a}\text{(t)and}\phantom{\rule{0.2em}{0ex}}{\overrightarrow{a}}^{\prime}\text{(t)}$ related?

The velocity of a particle in reference frame *A* is $(2.0\widehat{i}+3.0\widehat{j}\text{)}\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}.$ The velocity of reference frame *A* with respect to reference frame *B* is $4.0\widehat{k}\text{m}\text{/}\text{s},$ and the velocity of reference frame *B* with respect to *C* is $2.0\widehat{j}\text{m}\text{/}\text{s}.$ What is the velocity of the particle in reference frame *C*?

Raindrops fall vertically at 4.5 m/s relative to the earth. What does an observer in a car moving at 22.0 m/s in a straight line measure as the velocity of the raindrops?

A seagull can fly at a velocity of 9.00 m/s in still air. (a) If it takes the bird 20.0 min to travel 6.00 km straight into an oncoming wind, what is the velocity of the wind? (b) If the bird turns around and flies with the wind, how long will it take the bird to return 6.00 km?

A ship sets sail from Rotterdam, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction $40.0\text{\xb0}$ north of east. What is the velocity of the ship relative to Earth?

A boat can be rowed at 8.0 km/h in still water. (a) How much time is required to row 1.5 km downstream in a river moving 3.0 km/h relative to the shore? (b) How much time is required for the return trip? (c) In what direction must the boat be aimed to row straight across the river? (d) Suppose the river is 0.8 km wide. What is the velocity of the boat with respect to Earth and how much time is required to get to the opposite shore? (e) Suppose, instead, the boat is aimed straight across the river. How much time is required to get across and how far downstream is the boat when it reaches the opposite shore?

A small plane flies at 200 km/h in still air. If the wind blows directly out of the west at 50 km/h, (a) in what direction must the pilot head her plane to move directly north across land and (b) how long does it take her to reach a point 300 km directly north of her starting point?

A cyclist traveling southeast along a road at 15 km/h feels a wind blowing from the southwest at 25 km/h. To a stationary observer, what are the speed and direction of the wind?

A river is moving east at 4.0 m/s. A boat starts from the dock heading $30\text{\xb0}$ north of west at 7.0 m/s. If the river is 1800 m wide, (a) what is the velocity of the boat with respect to Earth and (b) how long does it take the boat to cross the river?