||| A spaceship maneuvering near Planet Zeta is located at $\vec{r}=(600î-400ĵ+200\widehat{k})\times {10}^3\text{km,}$ relative to the planet, and traveling at $\vec{v}=9500î\text{m/s}$. It turns on its thruster engine and accelerates with $\vec{a}=(40î-20\widehat{k})\text{m}/{\text{s}}^2$ for 35 min. What is the spaceship’s position when the engine shuts off? Give your answer as a position vector measured in km.

||| CALC A particle moving in the xy-plane has velocity $\vec{v}{}_0=v_{0x}î+v_{0y}ĵ$ at t $=$ 0. It undergoes acceleration $\vec{a}=btî-c{v}_yĵ$, where b and c are constants. Find an expression for the particle’s velocity at a later time t.

1. || A projectile is launched with speed $v_0$ and angle $\theta$. Derive an expression for the projectile’s maximum height h.

2. A baseball is hit with a speed of $33.6\text{m/s}$. Calculate its height and the distance traveled if it is hit at angles of $30.0\text{°}\text{,}45.0\text{°}$, and $60.0\text{°}$.

|| A projectile’s horizontal range over level ground is ${v_0}^2\text{sin}2\theta \text{/}g$. At what launch angle or angles will the projectile land at half of its maximum possible range?

|| CALC A projectile is launched from ground level at angle $\theta$ and speed $v_0$ into a headwind that causes a constant horizontal acceleration of magnitude a opposite the direction of motion.

1. Find an expression in terms of a and g for the launch angle that gives maximum range.

2. What is the angle for maximum range if a is 10% of g?

|| BIO A gray kangaroo can bound across level ground with each jump carrying it 10 m from the takeoff point. Typically the kangaroo leaves the ground at a $20\text{°}$ angle. If this is so:

1. What is its takeoff speed?

2. What is its maximum height above the ground?

|| A ball is thrown toward a cliff of height h with a speed of $30\text{m/s}$ and an angle of $60\text{°}$ above horizontal. It lands on the edge of the cliff 4.0 s later.

1. How high is the cliff?

2. What was the maximum height of the ball?

3. What is the ball’s impact speed?

|| You are target shooting using a toy gun that fires a small ball at a speed of 15 m/s. When the gun is fired at an angle of 30° above horizontal, the ball hits the bull’s-eye of a target at the same height as the gun. Then the target distance is halved. At what angle must you aim the gun to hit the bull’s-eye in its new position? (Mathematically there are two solutions to this problem; the physically reasonable answer is the smaller of the two.)

|| A tennis player hits a ball 2.0 m above the ground. The ball leaves his racquet with a speed of $20.0\text{m}\text{/}\text{s}$ at an angle $5.0\text{°}$ above the horizontal. The horizontal distance to the net is 7.0 m, and the net is 1.0 m high. Does the ball clear the net? If so, by how much? If not, by how much does it miss?

|| A snowboarder starts down a frictionless, 8.0-m-tall, 15° slope. The slope ends abruptly at the top of a 4.0-m-high wall that has level packed snow at its base. How far does the snowboarder land from the base of the wall?

|| You are watching an archery tournament when you start wondering how fast an arrow is shot from the bow. Remembering your physics, you ask one of the archers to shoot an arrow parallel to the ground. You find the arrow stuck in the ground 60 m away, making a $3.0\text{°}$ angle with the ground. How fast was the arrow shot?

|| You’re 6.0 m from one wall of the house seen in FIGURE P4.55. You want to toss a ball to your friend who is 6.0 m from the opposite wall. The throw and catch each occur 1.0 m above the ground.

1. What minimum speed will allow the ball to clear the roof?

2. At what angle should you toss the ball?

FIGURE P4.55

|| Sand moves without slipping at $6.0\text{m}\text{/}\text{s}$ down a conveyer that is tilted at $15\text{°}$. The sand enters a pipe 3.0 m below the end of the conveyer belt, as shown in FIGURE P4.56. What is the horizontal distance d between the conveyer belt and the pipe?

FIGURE P4.56

|| A stunt man drives a 1500 kg car at a speed of $20\text{m/s}$ off a 30-m-high cliff. The road leading to the cliff is inclined upward at an angle of $20\text{°}$.

1. How far from the base of the cliff does the car land?

2. What is the car’s impact speed?

||| BIO A javelin thrower standing at rest holds the center of the javelin behind her head, then accelerates it through a distance of 70 cm as she throws. She releases the 600 g javelin 2.0 m above the ground traveling at an angle of $30\text{°}$ above the horizontal. Top-rated javelin throwers do throw at about a $30\text{°}$ angle, not the $45\text{°}$ you might have expected, because the biomechanics of the arm allow them to throw the javelin much faster at $30\text{°}$ than they would be able to at $45\text{°}$. In this throw, the javelin hits the ground 62 m away. What was the acceleration of the javelin during the throw? Assume that it has a constant acceleration.

|| A cannonball is fired at 100 m/s from a barrel tilted upward at 25°.

1. Find an expression for the cannonball’s direction of travel, measured as an angle from horizontal, after traveling horizontal distance d.

2. How far has the cannonball traveled horizontally when it reaches its maximum height?

3. What is the angle after the cannonball travels 500 m?

|| Ships A and B leave port together. For the next two hours, ship A travels at 20 mph in a direction $30\text{°}$ west of north while ship B travels $20\text{°}$ east of north at 25 mph.

1. What is the distance between the two ships two hours after they depart?

2. What is the speed of ship A as seen by ship B?

|| While driving north at $25\text{m/s}$ during a rainstorm you notice that the rain makes an angle of $38\text{°}$ with the vertical. While driving back home moments later at the same speed but in the opposite direction, you see that the rain is falling straight down. From these observations, determine the speed and angle of the raindrops relative to the ground.

|| BIO You are asked to consult for the city’s research hospital, where a group of doctors is investigating the bombardment of cancer tumors with high-energy ions. As FIGURE P4.62 shows, ions are fired directly toward the center of the tumor at speeds of $5.0\times {10}^6\text{m}\text{/}\text{s}$. To cover the entire tumor area, the ions are deflected sideways by passing them between two charged metal plates that accelerate the ions perpendicular to the direction of their initial motion. The acceleration region is 5.0 cm long, and the ends of the acceleration plates are 1.5 m from the target. What sideways acceleration is required to deflect an ion 2.0 cm to one side?

FIGURE P4.62

|| CALC The angular velocity of a spinning gyroscope is measured every 0.5 s. The results and the best-fit line from a spreadsheet are shown in FIGURE P4.63.

1. What is the gyroscope’s initial angular velocity, at $t=0\text{s}$?

2. What is the angular acceleration at $t=2.0\text{s}$?

3. How many revolutions does the gyroscope make between $t=0\text{s}$ and $t=2.0\text{s}$?

FIGURE P4.63

|| A ball rolling on a circular track, starting from rest, has angular acceleration $\alpha$. Find an expression, in terms of $\alpha$, for the time at which the ball’s acceleration vector $\vec{a}$ is 45° away from a radial line toward the center of the circle.

|| A typical laboratory centrifuge rotates at 4000 rpm. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations.

1. What is the acceleration at the end of a test tube that is 10 cm from the axis of rotation?

2. For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of 1.0 m and stopped in a 1.0-ms-long encounter with a hard floor?

|| BIO Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge takes 30 s to speed up from rest to its top speed of 1 rotation every 1.3 s. The astronaut is strapped into a seat 6.0 m from the axis.

1. What is the astronaut’s tangential acceleration during the first 30 s?

2. How many g’s of acceleration does the astronaut experience when the device is rotating at top speed? Each $9.8{\text{m/s}}^2$ of acceleration is 1 g.

|| A Ferris wheel of radius R speeds up with angular acceleration $\alpha$ starting from rest. Find expressions for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle $\text{Δ}\theta$.

|| Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is $6.37\times {10}^6\text{m,}$ and the altitude of a geosynchronous orbit is $3.58\times {10}^7\text{m}$ $(\approx 22\text{,}000\text{miles})$. What are (a) the speed and (b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?

|| A computer hard disk 8.0 cm in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at $600\text{rad}\text{/}{\text{s}}^2$ for $\frac{1}{2}\text{s,}$ then coasts at a steady angular velocity for another $\frac{1}{2}\text{s}$.

1. What is the speed of the dot at $t=1.0\text{s}?$

2. Through how many revolutions has the disk turned?

|| A high-speed drill rotating ccw at 2400 rpm comes to a halt in 2.5 s.

1. What is the magnitude of the drill’s angular acceleration?

2. How many revolutions does it make as it stops?

|| Flywheels—rapidly rotating disks—are widely used in industry for storing energy. They are spun up slowly when extra energy is available, then decelerate quickly when needed to supply a boost of energy. A 20-cm-diameter rotor made of advanced materials can spin at 100,000 rpm.

1. What is the speed of a point on the rim of this rotor?

2. Suppose the rotor’s angular velocity decreases by 40% over 30 s as it supplies energy. What is the magnitude of the rotor’s angular acceleration? Assume that the angular acceleration is constant.

3. How many revolutions does the rotor make during these 30 s?

|| A 25 g steel ball is attached to the top of a 24-cm-diameter vertical wheel. Starting from rest, the wheel accelerates at $470\text{rad}\text{/}{\text{s}}^2$. The ball is released after $\frac{3}{4}$ of a revolution. How high does it go above the center of the wheel?

|| CALC The angular velocity of a process control motor is $\omega =\left(20-\frac{1}{2}{t}^2\right)\text{rad}\text{/}\text{s}$, where t is in seconds.

1. At what time does the motor reverse direction?

2. Through what angle does the motor turn between $t=0$ s and the instant at which it reverses direction?

|| CALC A 6.0-cm-diameter gear rotates with angular velocity $\omega =\left(20-\frac{1}{2}{t}^2\right)\text{rad}\text{/}\text{s}$, where t is in seconds. At $t=4.0$ s, what are:

1. The gear’s angular acceleration?

2. The tangential acceleration of a tooth on the gear?

|| CALC A painted tooth on a spinning gear has angular position $\theta =(6.0{\text{rad/s}}^4)t^4$. What is the tooth’s angular acceleration at the end of 10 revolutions?

||| A car starts from rest on a curve with a radius of 120 m and accelerates tangentially at $1.0{\text{m/s}}^2$. Through what angle will the car have traveled when the magnitude of its total acceleration is $2.0{\text{m/s}}^2?$

||| A long string is wrapped around a 6.0-cm-diameter cylinder, initially at rest, that is free to rotate on an axle. The string is then pulled with a constant acceleration of $1.5{\text{m/s}}^2$ until $1.0\text{m}$ of string has been unwound. If the string unwinds without slipping, what is the cylinder’s angular speed, in rpm, at this time?

$100\text{m}=0\text{m}+(50\text{c}\text{o}\text{s}\theta \text{m/s)}{t}_1$

$0\text{ m}=0\text{m}+(50\text{ sin }\theta \text{ m/s})t_1-\frac{1}{2}(9.80\text{ m/}{\text{s}}^2)t_1^2$

$v_x=-(6.0\text{cos}45\text{°})\text{m}\text{/}\text{s}+3.0\text{m}\text{/}\text{s}$

$v_y=(6.0\text{sin}45\text{°})\text{m}\text{/}\text{s}+0\text{m}\text{/}\text{s}$

$100\text{m}=v_y{t}_1\text{,}{x}_1=v_x{t}_1$

$2.5\text{rad}=0\text{rad}+{\omega }_i(10\text{s})+((1.5{\text{m/s}}^2)\text{/}2(50\text{m)})(10{\text{s)}}^2$

${\omega }_{\text{f}}={\omega }_i+((1.5{\text{m/s}}^2)\text{/}(50\text{m)})(10\text{s})$

||| A skateboarder starts up a 1.0-m-high, $30\text{°}$ ramp at a speed of $7.0\text{m}\text{/}\text{s}$. The skateboard wheels roll without friction. At the top she leaves the ramp and sails through the air. How far from the end of the ramp does the skateboarder touch down?

||| An archer standing on a $15\text{°}$ slope shoots an arrow $20\text{°}$ above the horizontal, as shown in FIGURE CP4.82. How far down the slope does the arrow hit if it is shot with a speed of $50\text{m/s}$ from 1.75 m above the ground?

FIGURE CP4.82

||| CALC The cannon in FIGURE CP4.83 fires a projectile at launch angle $\theta$ with respect to the slope, which is at angle $\varphi$. Find the launch angle that maximizes d.

FIGURE CP4.83

Hint: Choosing the proper coordinate system is essential. There are two options.

||| A cannon on a flat railroad car travels to the east with its barrel tilted 30° above horizontal. It fires a cannonball at 50 m/s. At $t=0\text{s}$, the car, starting from rest, begins to accelerate to the east at $2.0{\text{m/s}}^2$. At what time should the cannon be fired to hit a target on the tracks that is 400 m to the east of the car’s initial position? Assume that the cannonball is fired from ground level.

||| A child in danger of drowning in a river is being carried downstream by a current that flows uniformly with a speed of $2.0\text{m}\text{/}\text{s}$. The child is 200 m from the shore and 1500 m upstream of the boat dock from which the rescue team sets out. If the boat speed is 8.0 m/s with respect to the water, at what angle from the shore must the boat travel in order to reach the child?