If a cannonball is fired from a tall mountain, gravity

What exactly is a projectile?

Why does the vertical component of velocity for a projectile change with time, whereas the horizontal component of velocity doesn’t change?

A stone is thrown upward at an angle. What happens to the horizontal component of its velocity as it rises? As it falls?

A stone is thrown upward at an angle. What happens to the vertical component of its velocity as it rises? As it falls?

A projectile falls beneath the straight-line path it would follow if there were no gravity. How many meters does it fall below this line if it has been traveling for 1 s? For 2 s?

Do your answers to the preceding question depend on the angle at which the projectile is launched?

A projectile is launched vertically at 100 m/s. If air resistance can be ignored, at what speed will it return to its initial level?

A projectile is launched upward at an angle of \(70^{\circ}\) from the horizontal and strikes the ground a certain distance downrange. For what other angle of launch at the same speed would this projectile land just as far away?

How can a projectile “fall around the Earth”?

Why is it important that the projectile in the preceding question be above Earth’s atmosphere?

Why doesn’t the force of gravity change the speed of a bowling ball as it rolls along a bowling lane?

Why doesn’t the force of gravity change the speed of a satellite in circular orbit?

How much time does it take for a complete revolution of a satellite in close orbit about the Earth?

Why does the force of gravity change the speed of a satellite in an elliptical orbit?

For orbits of greater altitude, is the period longer or shorter?

At what part of an elliptical orbit does an Earth satellite have the greatest speed? The lowest speed?

Who gathered the data that showed planets traveling in elliptical paths around the Sun? Who discovered elliptical orbits? Who explained them?

What did Kepler discover about the periods of planets and their distances from the Sun?

What was the direction of the force on a planet in Kepler’s thinking? In Newton’s thinking?

Why is kinetic energy a constant for a satellite in a circular orbit but not for a satellite in an elliptical orbit?

Is the sum of kinetic and potential energies a constant for satellites in circular orbits, in elliptical orbits, or in both?

Escape speed from Earth is 11.2 km/s. Is it possible to escape from Earth at half this speed? At one-quarter this speed? If so, how?

A ball is thrown horizontally from a cliff at a speed of 10 m/s. You predict that its speed 1 s later will be slightly greater than 14 m/s. Your friend says it will be 10 m/s. Show who is correct.

You’re in an airplane that flies horizontally with speed 1000 km/h (280 m/s) when an engine falls off. Ignore air resistance and assume it takes 30 s for the engine to hit the ground. (a) Show that the airplane is 4.5 km high. (b) Show that the horizontal distance that the aircraft engine moves during its fall is 8400 m. (c) If the airplane somehow continues to fly as if nothing had happened, where is the engine relative to the airplane at the moment the engine hits the ground?

A baseball projected with an initial velocity of 141 m/s at an angle of \(45^{\circ}\) follows a parabolic path and hits a balloon at the top of its trajectory. Ignoring air resistance, show that the ball hits the balloon at a speed of 100 m/s. Text Transcription: 45^circ

Students in the lab (see Figure 10.5) measure the speed of a steel ball to be 8.0 m/s when launched horizontally from a 1.0-m-high tabletop. Their objective is to place a 20-cm-tall coffee can on the floor to catch the ball. Show that they score a bull’s-eye when the can is placed 3.2 m from the base of the table.

At a particular point in its orbit, a satellite in an elliptical orbit has a gravitational potential energy of 5000 MJ with respect to Earth’s surface and a kinetic energy of 4500 MJ. Later in its orbit, the satellite’s potential energy is 6000 MJ. What is its kinetic energy at that point?

A certain satellite has a kinetic energy of 8 billion joules at perigee (the point closest to Earth) and 5 billion joules at apogee (the point farthest from Earth). As the satellite travels from apogee to perigee, how much work does the gravitational force do on it? Does its potential energy increase or decrease during this time, and by how much?

Calculate the hang time of a person who moves 3 m horizontally during a 1.25-m high jump. What is the hang time when the person moves 6 m horizontally during this jump?

A horizontally moving tennis ball barely clears the net, a distance \(y\) above the surface of the court. To land within the tennis court, the ball must not be moving too fast. (a) To remain within the court’s border, a horizontal distance \(d\) from the bottom of the net, show that the ball’s maximum speed over the net is \(v=\frac{d}{\sqrt{\frac{2 y}{g}}}\) (b) Suppose the height of the net is \(1.00 \ \mathrm{m}\), and the court’s border is \(12.0 \ \mathrm{m}\) from the bottom of the net. Use \(g=10 \mathrm{m} / \mathrm{s}^{2}\) and show that the maximum speed of the horizontally moving ball clearing the net is about \(27 \ \mathrm{m}/\mathrm{s}\) (about \(60 \ \mathrm{mi}/\mathrm{h}\)). (c) Does the mass of the ball make a difference? Defend your answer. Equation Transcription: y d 1.00 m 12.0 m g = 10 m/s2 27 m/s 60 mi/h Text Transcription: y d v = d/sqrt 2y/g 1.00 m 12.0 m g = 10 m/s^2 27 m/s 60 mi/h

A ball is thrown upward at the velocities and angles shown. From greatest to least, rank them by their (a) vertical components of velocity. (b) horizontal components of velocity. (c) accelerations when they reach the top of their paths.

A ball is tossed off the edge of a cliff with the same speed but at different angles, as shown. From greatest to least, rank the (a) initial PEs of the balls relative to the ground below. (b) initial KEs of the balls when tossed. (c) KEs of the balls when they hit the ground below. (d) times of flight while airborne.

The dashed lines show three circular orbits about Earth. Rank the following quantities from greatest to least: (a) Their orbital speed (b) Their time to orbit Earth

The positions of a satellite in elliptical orbit are indicated. Rank these quantities from greatest to least: a) Gravitational force (b) Speed (c) Momentum (d) KE (e) PE (f) Total energy (KE + PE) (g) Acceleration

In synchronized diving, divers remain in the air for the same time. With no air drag, they would fall together. But air drag is appreciable, and divers may differ significantly in size and weight, so how do they remain together in the fall?

Suppose you roll a ball off a tabletop. Will the time to hit the floor depend on the speed of the ball? (Will a fast ball take a longer time to hit the floor?) Defend your answer.

Suppose you roll a ball off a tabletop. Compared with a slow roll, will a faster-moving ball hit the floor with a higher speed? Defend your answer.

If you toss a ball vertically upward in a uniformly moving train, it returns to its starting place. Will it do the same if the train is accelerating? Explain.

Suppose you drop a care package from an airplane traveling at constant velocity, and further suppose that air resistance doesn’t affect the falling package. What will be its falling path as observed by someone at rest on the ground, not directly below but off to the side where there’s a clear view? What will be the falling path as observed by you looking downward from the airplane? Where will the package strike the ground, relative to you in the airplane? Where will it strike in the more realistic case in which air resistance does affect the fall?

A heavy crate accidentally falls from a high-flying airplane just as it flies directly above a shiny red Porsche smartly parked in a car lot. Relative to the Porsche, where will the crate crash?

Fragments of fireworks beautifully illuminate the night sky. (a) What specific path is ideally traced by each fragment? (b) What paths would the fragments trace in a gravity-free region?

In the absence of air resistance, why doesn’t the horizontal component of a projectile’s motion change, while the vertical component does?

If air drag can be ignored, at what point in its trajectory does a batted baseball have its minimum speed? How does this compare with the horizontal component of its velocity at other points?

When a rifle is being fired at a distant target, why isn’t the barrel aligned so that it points exactly at the target?

Two golfers each hit a ball at the same speed, but one at \(60^{\circ}\) with the horizontal and the other at \(30^{\circ}\). Which ball goes farther? Which hits the ground first? (Ignore air resistance.)

A park ranger shoots a monkey hanging from a branch of a tree with a tranquilizing dart. The ranger aims directly at the monkey, not realizing that the dart will follow a parabolic path and thus will fall below the monkey. The monkey, however, sees the dart leave the gun and lets go of the branch to avoid being hit. Will the monkey be hit anyway? Does the velocity of the dart affect your answer, assuming that it is great enough to travel the horizontal distance to the tree before hitting the ground? Defend your answer.

A projectile is fired straight upward at 141 m/s. How fast is it moving at the instant it reaches the top of its trajectory? Suppose that it were fired upward at \(45^{\circ}\) instead. Then its horizontal component of velocity would be 100 m/s. What would be the speed of the projectile at the top of its trajectory? Text Transcription: 45^circ

When you jump upward, your hang time is the time your feet are off the ground. Does hang time depend on your vertical component of velocity when you jump, your horizontal component of velocity, or both? Defend your answer.

The hang time of a basketball player who jumps a vertical distance of 2 feet (0.6 m) is about \(\frac{2}{3}\) second. What will be the hang time if the player reaches the same height while jumping 4 feet (1.2 m) horizontally?

Since the Moon is gravitationally attracted to Earth, why doesn’t it simply crash into Earth?

Which planets have a more-than-one-Earth-year period: planets nearer than Earth to the Sun, or planets farther from the Sun than Earth?

When a satellite coasts in a circular orbit at constant speed about Earth, is it accelerating? If so, in what direction? If not, why not?

Does the speed of a falling object depend on its mass? Does the speed of a satellite in orbit depend on its mass? Defend your answers.

Upon which does the speed of a circling satellite not depend: the mass of the satellite, the mass of Earth, the distance of the satellite from Earth?

A circularly moving object requires a centripetal force. What supplies this force for satellites that orbit Earth?

If you have ever watched the launching of an Earth satellite, you may have noticed that the rocket starts vertically upward, then departs from a vertical course and continues its climb at an angle. Why does it start vertically? Why doesn’t it continue vertically?

If a cannonball is fired from a tall mountain, gravity changes its speed all along its trajectory. But if it is fired fast enough to go into circular orbit, gravity does not change its speed at all. Explain.

When a satellite in circular orbit slows, perhaps due to the firing of a “retro rocket,” it ends up gaining more speed than it had initially. Why?

Earth is closer to the Sun in December than in June. In which of these two months is Earth moving faster around the Sun?

Ignoring air resistance, could a satellite be put into orbit in a circular tunnel beneath Earth’s surface? Discuss.

Why is work done by the force of gravity on a satellite when it moves from one part of an elliptical orbit to another, but no work is involved when it moves from one part of a circular orbit to another?

In the sketch on the left, a ball gains \(KE\) when rolling down a hill because work is done by the component of weight \((F)\) that acts in the direction of motion. Sketch in the similar component of gravitational force that does work to change the \(KE\) of the satellite on the right. Equation Transcription: Text Transcription: KE F KE

What is the shape of the orbit when the velocity of the satellite is everywhere perpendicular to the force of gravity?

If a satellite circled Earth at a distance equal to the Earth–Moon distance, how long would it take for it to make a complete orbit? In other words, what would be its period?

Can a satellite coast in a stable orbit in a plane that doesn’t intersect Earth’s center? Defend your answer.

What happens to an Earth satellite’s period when the satellite is placed into a higher orbit?

If a flight mechanic drops a box of tools from a high-flying jumbo jet, the box crashes to Earth. An astronaut on the orbiting space shuttle in 2008 accidentally dropped a box of tools. Why didn’t the tools crash to Earth? Defend your answer.

How could an astronaut in a space shuttle “drop” an object vertically to Earth?

A high-orbiting spaceship travels at 7 km/s with respect to Earth. Suppose it projects a capsule rearward at 7 km/s with respect to the ship. Describe the path of the capsule with respect to Earth.

A satellite in a circular orbit about the Moon fires a small probe in a direction opposite to the velocity of the satellite. If the speed of the probe relative to the satellite is the same as the satellite’s speed relative to the Moon, describe the motion of the probe. If the probe’s relative speed is twice the speed of the satellite, why would it pose a danger to the satellite?

The orbital velocity of Earth about the Sun is 30 km/s. If Earth were suddenly stopped in its tracks, it would simply fall radially into the Sun. A congressional leader suggests firing a rocket loaded with radioactive wastes into the Sun for permanent disposal. For this to occur, with what speed and in what direction with respect to Earth’s orbit should the rocket be fired?

If you stopped an Earth satellite dead in its tracks, it would simply crash into Earth. Why, then, don’t the communications satellites that “hover motionless” above the same spot on Earth crash into Earth?

What is the advantage of launching space vehicles from high-flying aircraft instead of from the ground?

Which requires less fuel: launching a rocket to escape speed from the Moon or from Earth? Defend your answer.

What is the maximum possible speed of impact on the surface of Earth for a faraway body initially at rest that falls to Earth by virtue of Earth’s gravity only?

As part of their training before going into orbit, astronauts experience weightlessness when riding in an airplane that is flown along the same parabolic trajectory as a freely falling projectile. A classmate says that gravitational forces on everything inside the plane during this maneuver cancel to zero. Another classmate looks to you for confirmation. What is your response?

At which of the indicated positions does the satellite in an elliptical orbit experience the greatest gravitational force? Have the greatest speed? The greatest velocity? The greatest momentum? The greatest kinetic energy? The greatest gravitational potential energy? The greatest total energy? The greatest angular momentum? The greatest acceleration?

At what point in its elliptical orbit about the Sun is the acceleration of Earth toward the Sun a maximum? At what point is it a minimum? Defend your answers.

For maximum range, a football should be punted at about \(45^{\circ}\) to the horizontal—somewhat less due to air drag. But punts are often kicked at angles greater than \(45^{\circ}\). Discuss why this is so.

A friend claims that bullets fired by some high-powered rifles travel for many meters in a straight-line path before they start to fall. Another friend disputes this claim and states that all bullets from any rifle drop beneath a straight-line path a vertical distance given by \(\frac{1}{2} g t^{2}\) and that the curved path is apparent for low velocities and less apparent for high velocities. Now it’s your turn: Discuss whether or not all fired bullets drop the same vertical distance in equal times. Equation Transcription: Text Transcription: 1/2 gt^2

Mars has about \(\frac{1}{9}\) the mass of Earth. If Mars were somehow positioned into the same orbit as Earth’s, discuss how its time to circle the Sun would compare with Earth’s. (Longer, shorter, or the same?)

Would the speed of a satellite in close circular orbit about Jupiter be greater than, equal to, or less than 8 km/s?

Discuss why Hawaii is the most efficient launching site in the United States for nonpolar satellites. In your discussion include a look at the spinning Earth from above either pole and compare it to a rotating turntable.

Why are satellites normally sent into orbit by firing them in an easterly direction, the direction in which Earth spins? And discuss why locations near the equator are favored.

Discuss why a satellite can orbit at 5 km above the Moon, but not at 5 km above Earth.

A communications satellite with a 24-hour period hovers over a fixed point on Earth and, to Earth observers, seems motionless. Why is it placed only in an orbit in the plane of Earth’s equator? In your discussion, think of the satellite’s orbit as a ring around Earth.

A “geosynchronous” Earth satellite can remain nearly directly overhead in Singapore but not in San Francisco. Discuss why this is so.

In 2000–2001, NASA’s Near Earth Asteroid Rendezvous (NEAR) spacecraft orbited around the 20-mile-long asteroid Eros. Do you speculate that the orbital speed of this spacecraft is greater or less than 8 km/s? Discuss and defend your answer.

Why does a satellite burn up when it descends into the atmosphere? And why doesn’t it burn up when it ascends through the atmosphere?

Which two planets are never seen at midnight, and why?

Can a satellite maintain an orbit in the plane of the Arctic Circle? Why or why not?

You read in an article about astronauts in a major magazine that “about 62 miles up, the atmosphere ends and gravity becomes very weak. . . .” What error is made here?

Discuss the requirement that a GPS device needs information from at least three GPS satellites and why one “very good” satellite cannot do the task?

In an accidental explosion, a satellite breaks in half while in circular orbit about Earth. One half is brought momentarily to rest. What is the fate of the half brought to rest? Discuss what happens to the other half.

A giant rotating wheel in space provides artificial gravity for its occupants, as discussed in Chapter 8. Instead of a full wheel, discuss the idea of a pair of capsules joined by a tether line and rotating about each other. Can such an arrangement provide artificial gravity for the occupants?

A rocket coasts in an elliptical orbit around Earth. To attain the greatest amount of KE for escape for a given amount of fuel, should it fire its engines to accelerate forward when it is at the apogee or at the perigee? (Hint: Let the formula \(F d=\Delta \mathrm{KE}\) be your guide to thinking. Suppose the thrust F is brief and of the same duration in either case. Then consider the distance d the rocket would travel during this brief burst at the apogee and at the perigee.)