Problems

The position of a particle changes from ${\overrightarrow{r}}_1=(2.0\text{}\widehat{i}+3.0\widehat{j})\text{cm}$ to ${\overrightarrow{r}}_2=(\mathrm{-4.0}\widehat{i}+3.0\widehat{j})\text{cm}.$ What is the particle’s displacement?

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?

New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and $45\text{°}$ 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.

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 $45\text{°}$ with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at $90\text{°}$ 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 position of a particle is $\overrightarrow{r}(t)=(3.0t^2\widehat{i}+5.0\widehat{j}-6.0t\widehat{k})\text{m}.$ (a) Determine its velocity and acceleration as functions of time. (b) What are its velocity and acceleration at time t = 0?

A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of $2.0\text{m/}{\text{s}}^2\widehat{i}.$ A strong wind is pushing the boat, giving it an additional velocity of $2.0\text{m/s}\widehat{i}+1.0\text{m/s}\widehat{j}.$ (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 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 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 $5.0\text{m/}{\text{s}}^2$ at $30\text{°}$ 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?

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?

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?

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 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 $60\text{°}$ above the horizontal, what must the initial speed be if it were to go through the basket?

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.

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?

Trying to escape his pursuers, a secret agent skis off a slope inclined at $30\text{°}$ 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 projectile is shot at a hill, the base of which is 300 m away. The projectile is shot at $60\text{°}$ above the horizontal with an initial speed of 75 m/s. The hill can be approximated by a plane sloped at $20\text{°}$ to the horizontal. Relative to the coordinate system shown in the following figure, the equation of this straight line is $y=(\text{tan}20\text{°})x-109.$ Where on the hill does the projectile land?

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 $15\text{°},$ what was his initial speed?

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 $40\text{°}$ 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?

The Lunar Roving Vehicle used in NASA’s late Apollo missions reached an unofficial lunar land speed of $5.0\text{m/s}$ 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 $20\text{°},$ how long would it be “airborne” on the Moon?

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 $3.7\text{m}\text{/}{\text{s}}^2.$

You throw a baseball at an initial speed of 15.0 m/s at an angle of $30\text{°}$ with respect to the horizontal. What would the ball’s initial speed have to be at $30\text{°}$ 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.

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?

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 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 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?

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 ${\overrightarrow{v}}_{S^{\prime }S}=(1.0\widehat{i}+2.0\widehat{j}+3.0\widehat{k})t\text{m}\text{/}\text{s}$. (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 $\overrightarrow{v}(t)\text{and}{\overrightarrow{v}}^{\prime }(t)?$ (d) How are accelerations $\overrightarrow{a}\text{(t) and}{\overrightarrow{a}}^{\prime }\text{(t)}$ related?

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 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 $40.0\text{°}$ north of east. What is the velocity of the ship relative to Earth?

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 river is moving east at 4.0 m/s. A boat starts from the dock heading $30\text{°}$ 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?