### Problems

## 9.1 Linear Momentum

An elephant and a hunter are having a confrontation.

- Calculate the momentum of the 2000.0-kg elephant charging the hunter at a speed of 7.50 m/s.
- Calculate the ratio of the elephant’s momentum to the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s.
- What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?

A skater of mass 40 kg is carrying a box of mass 5 kg. The skater has a speed of 5 m/s with respect to the floor and is gliding without any friction on a smooth surface.

- Find the momentum of the box with respect to the floor.
- Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface.

A car of mass 2000 kg is moving with a constant velocity of 10 m/s due east. What is the momentum of the car?

The mass of Earth is $5.97\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{24}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and its orbital radius is an average of $1.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{11}\phantom{\rule{0.2em}{0ex}}\text{m}$. Calculate the magnitude of its linear momentum at the location in the diagram.

If a rainstorm drops 1 cm of rain over an area of 10 km^{2} in the period of 1 hour, what is the momentum of the rain that falls in one second? Assume the terminal velocity of a raindrop is 10 m/s.

What is the average momentum of an avalanche that moves a 40-cm-thick layer of snow over an area of 100 m by 500 m over a distance of 1 km down a hill in 5.5 s? Assume a density of 350 kg/m^{3} for the snow.

What is the average momentum of a 70.0-kg sprinter who runs the 100-m dash in 9.65 s?

## 9.2 Impulse and Collisions

A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment (see the following figure).

- Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm.
- Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm.

One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of $4.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{m/s}$, given the collision lasts $6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\phantom{\rule{0.2em}{0ex}}\text{s}$.

A cruise ship with a mass of $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{kg}$ strikes a pier at a speed of 0.750 m/s. It comes to rest after traveling 6.00 m, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (*Hint*: First calculate the time it took to bring the ship to rest, assuming a constant force.)

Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of $1.76\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{N}$ for $5.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\phantom{\rule{0.2em}{0ex}}\text{s}$.

Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.

A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. Assume constant acceleration of the hammer-nail pair.

- Calculate the duration of the impact.
- What was the average force exerted on the nail?

What is the momentum (as a function of time) of a 5.0-kg particle moving with a velocity $\overrightarrow{v}(t)=\left(2.0\widehat{i}+4.0t\widehat{j}\right)\phantom{\rule{0.2em}{0ex}}\text{m/s}$? What is the net force acting on this particle?

The *x*-component of a force on a 46-g golf ball by a 7-iron versus time is plotted in the following figure:

- Find the
*x*-component of the impulse during the intervals- [0, 50 ms], and
- [50 ms, 100 ms]

- Find the change in the
*x*-component of the momentum during the intervals- [0, 50 ms], and
- [50 ms, 100 ms]

A hockey puck of mass 150 g is sliding due east on a frictionless table with a speed of 10 m/s. Suddenly, a constant force of magnitude 5 N and direction due north is applied to the puck for 1.5 s. Find the north and east components of the momentum at the end of the 1.5-s interval.

A ball of mass 250 g is thrown with an initial velocity of 25 m/s at an angle of $30\text{\xb0}$ with the horizontal direction. Ignore air resistance. What is the momentum of the ball after 0.2 s? (Do this problem by finding the components of the momentum first, and then constructing the magnitude and direction of the momentum vector from the components.)

## 9.3 Conservation of Linear Momentum

Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of $1.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and a velocity of $(0.30\phantom{\rule{0.2em}{0ex}}\text{m/s})\widehat{i}$, and the second having a mass of $1.10\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and a velocity of $\text{\u2212}(0.12\phantom{\rule{0.2em}{0ex}}\text{m/s})\widehat{i}$. What is their final velocity?

Two identical pucks collide elastically on an air hockey table. Puck 1 was originally at rest; puck 2 has an incoming speed of 6.00 m/s and scatters at an angle of $30\text{\xb0}$ with respect to its incoming direction. What is the velocity (magnitude and direction) of puck 1 after the collision?

The figure below shows a bullet of mass 200 g traveling horizontally towards the east with speed 400 m/s, which strikes a block of mass 1.5 kg that is initially at rest on a frictionless table.

After striking the block, the bullet is embedded in the block and the block and the bullet move together as one unit.

- What is the magnitude and direction of the velocity of the block/bullet combination immediately after the impact?
- What is the magnitude and direction of the impulse by the block on the bullet?
- What is the magnitude and direction of the impulse from the bullet on the block?
- If it took 3 ms for the bullet to change the speed from 400 m/s to the final speed after impact, what is the average force between the block and the bullet during this time?

A 20-kg child is coasting at 3.3 m/s over flat ground in a 4.0-kg wagon.

- If the child drops a 1-kg ball out the back of the wagon, what is the final speed of the child and wagon?
- If the child drops a 1-kg ball out the back of the wagon so that the ball is at rest relative to the ground, what is the final speed of the child and wagon?

A 4.5 kg puffer fish expands to 40% of its mass by taking in water. When the puffer fish is threatened, it releases the water toward the threat to move quickly forward. What is the ratio of the speed of the puffer fish forward to the speed of the expelled water backwards?

Explain why a cannon recoils when it fires a shell.

Two figure skaters are coasting in the same direction, with the leading skater moving at 5.5 m/s and the trailing skating moving at 6.2 m/s. When the trailing skater catches up with the leading skater, he picks her up without applying any horizontal forces on his skates. If the trailing skater is 50% heavier than the 50-kg leading skater, what is their speed after he picks her up?

A 2000-kg railway freight car coasts at 4.4 m/s underneath a grain terminal, which dumps grain directly down into the freight car. If the speed of the loaded freight car must not go below 3.0 m/s, what is the maximum mass of grain that it can accept?

## 9.4 Types of Collisions

A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of 15.8° to the initial direction of the bowling ball and with a speed of 15.0 m/s.

- Calculate the final velocity (magnitude and direction) of the bowling ball.
- Is the collision elastic?

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei. The energy of the incoming helium nucleus was $8.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-13}}\phantom{\rule{0.2em}{0ex}}\text{J}$, and the masses of the helium and gold nuclei were $6.68\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-27}}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and $3.29\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-25}}\phantom{\rule{0.2em}{0ex}}\text{kg}$, respectively (note that their mass ratio is 4 to 197).

a. If a helium nucleus scatters to an angle of $120\text{\xb0}$ during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus.

b. What is the final kinetic energy of the helium nucleus?

A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?

A 100-g firecracker is launched vertically into the air and explodes into two pieces at the peak of its trajectory. If a 72-g piece is projected horizontally to the left at 20 m/s, what is the speed and direction of the other piece?

In an elastic collision, a 400-kg bumper car collides directly from behind with a second, identical bumper car that is traveling in the same direction. The initial speed of the leading bumper car is 5.60 m/s and that of the trailing car is 6.00 m/s. Assuming that the mass of the drivers is much, much less than that of the bumper cars, what are their final speeds?

Repeat the preceding problem if the mass of the leading bumper car is 30.0% greater than that of the trailing bumper car.

An alpha particle (^{4}He) undergoes an elastic collision with a stationary uranium nucleus (^{235}U). What percent of the kinetic energy of the alpha particle is transferred to the uranium nucleus? Assume the collision is one-dimensional.

You are standing on a very slippery icy surface and throw a 1-kg football horizontally at a speed of 6.7 m/s. What is your velocity when you release the football? Assume your mass is 65 kg.

A 35-kg child rides a relatively massless sled down a hill and then coasts along the flat section at the bottom, where a second 35-kg child jumps on the sled as it passes by her. If the speed of the sled is 3.5 m/s before the second child jumps on, what is its speed after she jumps on?

A boy sleds down a hill and onto a frictionless ice-covered lake at 10.0 m/s. In the middle of the lake is a 1000-kg boulder. When the sled crashes into the boulder, he is propelled backwards from the boulder. The collision is an elastic collision. If the boy’s mass is 40.0 kg and the sled’s mass is 2.50 kg, what is the speed of the sled and the boulder after the collision?

## 9.5 Collisions in Multiple Dimensions

A 0.90-kg falcon is diving at 28.0 m/s at a downward angle of $35\text{\xb0}$. It catches a 0.325-kg pigeon from behind in midair. What is their combined velocity after impact if the pigeon’s initial velocity was 7.00 m/s directed horizontally? Note that ${\widehat{v}}_{\text{1,}\text{i}}$ is a unit vector pointing in the direction in which the falcon is initially flying.

A billiard ball, labeled 1, moving horizontally strikes another billiard ball, labeled 2, at rest. Before impact, ball 1 was moving at a speed of 3.00 m/s, and after impact it is moving at 0.50 m/s at 50° from the original direction. If the two balls have equal masses of 300 g, what is the velocity of the ball 2 after the impact?

A projectile of mass 2.0 kg is ﬁred in the air at an angle of 40.0$\text{\xb0}$ to the horizon at a speed of 50.0 m/s. At the highest point in its flight, the projectile breaks into three parts of mass 1.0 kg, 0.7 kg, and 0.3 kg. The 1.0-kg part falls straight down after breakup with an initial speed of 10.0 m/s, the 0.7-kg part moves in the original forward direction, and the 0.3-kg part goes straight up.

- Find the speeds of the 0.3-kg and 0.7-kg pieces immediately after the break-up.
- How high from the break-up point does the 0.3-kg piece go before coming to rest?
- Where does the 0.7-kg piece land relative to where it was ﬁred from?

Two asteroids collide and stick together. The first asteroid has mass of $15\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and is initially moving at 770 m/s. The second asteroid has mass of $20\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and is moving at 1020 m/s. Their initial velocities made an angle of 20° with respect to each other. What is the final speed and direction with respect to the velocity of the first asteroid?

A 200-kg rocket in deep space moves with a velocity of $\left(121\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}+\left(38.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$. Suddenly, it explodes into three pieces, with the first (78 kg) moving at $\text{\u2212}\left(321\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}+\left(228\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$ and the second (56 kg) moving at $\left(16.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}-\left(88.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$. Find the velocity of the third piece.

A proton traveling at $3.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ scatters elastically from an initially stationary alpha particle and is deflected at an angle of 85° with respect to its initial velocity. Given that the alpha particle has four times the mass of the proton, what percent of its initial kinetic energy does the proton retain after the collision?

Three 70-kg deer are standing on a flat 200-kg rock that is on an ice-covered pond. A gunshot goes off and the deer scatter, with deer A running at $\left(15\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}+\left(5.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$, deer B running at $\left(\mathrm{-12}\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}+\left(8.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$, and deer C running at $\left(1.2\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{i}-\left(18.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\widehat{j}$. What is the velocity of the rock on which they were standing?

A family is skating. The father (75 kg) skates at 8.2 m/s and collides and sticks to the mother (50 kg), who was initially moving at 3.3 m/s and at 45° with respect to the father’s velocity. The pair then collides with their daughter (30 kg), who was stationary, and the three slide off together. What is their final velocity?

An oxygen atom (mass 16 u) moving at 733 m/s at 15.0° with respect to the $\widehat{i}$ direction collides and sticks to an oxygen molecule (mass 32 u) moving at 528 m/s at 128° with respect to the $\widehat{i}$ direction. The two stick together to form ozone. What is the final velocity of the ozone molecule?

Two cars of the same mass approach an extremely icy four-way perpendicular intersection. Car A travels northward at 30 m/s and car B is travelling eastward. They collide and stick together, traveling at 28° north of east. What was the initial velocity of car B?

## 9.6 Center of Mass

Three point masses are placed at the corners of a triangle as shown in the figure below.

Find the center of mass of the three-mass system.

Two particles of masses ${m}_{1}$ and ${m}_{2}$ separated by a horizontal distance *D* are released from the same height *h* at the same time. Find the vertical position of the center of mass of these two particles at a time before the two particles strike the ground. Assume no air resistance.

Two particles of masses ${m}_{1}$ and ${m}_{2}$ separated by a horizontal distance *D* are let go from the same height *h* at different times. Particle 1 starts at $t=0$, and particle 2 is let go at $t=T$. Find the vertical position of the center of mass at a time before the first particle strikes the ground. Assume no air resistance.

Two particles of masses ${m}_{1}$ and ${m}_{2}$ move uniformly in different circles of radii ${R}_{1}$ and ${R}_{2}$ about origin in the *x*, *y*-plane. The *x*- and *y*-coordinates of the center of mass and that of particle 1 are given as follows (where length is in meters and *t* in seconds):

${x}_{1}(t)=4\text{cos}(2t),{y}_{1}(t)=4\text{sin}(2t)$

and:

${x}_{\text{CM}}(t)=3\text{cos}(2t),{y}_{\text{CM}}(t)=3\text{sin}(2t).$

- Find the radius of the circle in which particle 1 moves.
- Find the
*x*- and*y*-coordinates of particle 2 and the radius of the circle this particle moves.

Two particles of masses ${m}_{1}$ and ${m}_{2}$ move uniformly in different circles of radii ${R}_{1}$ and${R}_{2}$ about the origin in the *x*, *y*-plane. The coordinates of the two particles in meters are given as follows ($z=0$ for both). Here *t* is in seconds:

$\begin{array}{ccc}\hfill {x}_{1}(t)& =\hfill & 4\phantom{\rule{0.2em}{0ex}}\text{cos}(2t)\hfill \\ \hfill {y}_{1}(t)& =\hfill & 4\phantom{\rule{0.2em}{0ex}}\text{sin}(2t)\hfill \\ \hfill {x}_{2}(t)& =\hfill & 2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(3t-\frac{\pi}{2}\right)\hfill \\ \hfill {y}_{2}(t)& =\hfill & 2\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3t-\frac{\pi}{2}\right)\hfill \end{array}$

- Find the radii of the circles of motion of both particles.
- Find the
*x*- and*y*-coordinates of the center of mass. - Decide if the center of mass moves in a circle by plotting its trajectory.

Find the center of mass of a one-meter long rod, made of 50 cm of iron (density $8\phantom{\rule{0.2em}{0ex}}\frac{\text{g}}{{\text{cm}}^{3}}$) and 50 cm of aluminum (density $2.7\phantom{\rule{0.2em}{0ex}}\frac{\text{g}}{{\text{cm}}^{3}}$).

Find the center of mass of a rod of length *L* whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the *x*-axis with one end at the origin and the other end at $x=L$, the density is given by $\rho (x)={\rho}_{0}+\left({\rho}_{1}-{\rho}_{0}\right){\left(\frac{x}{L}\right)}^{2}$, where ${\rho}_{0}$ and ${\rho}_{1}$ are constant values.

Find the center of mass of a rectangular block of length *a* and width *b* that has a nonuniform density such that when the rectangle is placed in the *x*,*y*-plane with one corner at the origin and the block placed in the first quadrant with the two edges along the *x*- and *y*-axes, the density is given by$\rho (x,y)={\rho}_{0}x$, where ${\rho}_{0}$ is a constant.

Find the center of mass of a rectangular material of length *a* and width *b* made up of a material of nonuniform density. The density is such that when the rectangle is placed in the *xy*-plane, the density is given by $\rho (x,y)={\rho}_{0}xy$.

A cube of side *a* is cut out of another cube of side *b* as shown in the figure below.

Find the location of the center of mass of the structure. (*Hint:* Think of the missing part as a negative mass overlapping a positive mass.)

Find the center of mass of a cone of uniform density that has a radius *R* at the base, height *h*, and mass *M*. Let the origin be at the center of the base of the cone and have +*z* going through the cone vertex.

Find the center of mass of a thin wire of mass *m* and length *L* bent in a semicircular shape. Let the origin be at the center of the semicircle and have the wire arc from the +*x* axis, cross the +*y* axis, and terminate at the −*x* axis.

Find the center of mass of a uniform thin semicircular plate of radius *R*. Let the origin be at the center of the semicircle, the plate arc from the +*x* axis to the −x axis, and the *z* axis be perpendicular to the plate.

Find the center of mass of a sphere of mass *M* and radius *R* and a cylinder of mass *m*, radius *r*, and height *h* arranged as shown below.

Express your answers in a coordinate system that has the origin at the center of the cylinder.

## 9.7 Rocket Propulsion

(a) A 5.00-kg squid initially at rest ejects 0.250 kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-*N* frictional force opposing the squid’s movement?

(b) How much energy is lost to work done against friction?

A rocket takes off from Earth and reaches a speed of 100 m/s in 10.0 s. If the exhaust speed is 1500 m/s and the mass of fuel burned is 100 kg, what was the initial mass of the rocket?

Repeat the preceding problem but for a rocket that takes off from a space station, where there is no gravity other than the negligible gravity due to the space station.

How much fuel would be needed for a 1000-kg rocket (this is its mass with no fuel) to take off from Earth and reach 1000 m/s in 30 s? The exhaust speed is 1000 m/s.

What exhaust speed is required to accelerate a rocket in deep space from 800 m/s to 1000 m/s in 5.0 s if the total rocket mass is 1200 kg and the rocket only has 50 kg of fuel left?

**Unreasonable Results** Squids have been reported to jump from the ocean and travel 30.0 m (measured horizontally) before re-entering the water.

(a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0°, assuming negligible lift from the air and negligible air resistance.

(b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at 12.0 m/s; gravitational force and friction are neglected.

(c) What is unreasonable about the results?

(d) Which premise is unreasonable, or which premises are inconsistent?