Chapter 10 Projectile Motion Physics

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?

5 m, 20 m. From d = ½gt²

A girl throws a ball horizontally. As the ball leaves the girl's hand, 1 second later it will have fallen:

5 meters below the horizontal component. Whatever the speed, the ball will fall a vertical distance of 5 meters below the horizontal component.

Neglecting air drag, a ball tossed at an angle of 30° with the horizontal will go as far downrange as one that is tossed at the same speed at an angle of what?

60°. Horizontal distances will be the same for launching angles that add up to 90°. So 90-30 =60

What speed is required to obtain a circular orbit around the Earth?

8 km/s.

At what part of its trajectory does the baseball have a minimum speed?

A ball's minimum speed occurs at the top of its trajectory. If it is launched vertically, its speed at the top is zero. If launched at an angle, the vertical component of velocity is zero at the top, leaving only the horizontal component. So the speed at the top is equal to the horizontal component of the ball's velocity at any point.

What exactly is a projectile?

A projectile is any object is projected by some means and continues in motion by its own inertia.

Geosynchronous orbit

A satellite orbit in which the satellite orbits Earth once each day. When moving westward, the satellite remains at a fixed point - about 36,000 k, above earth's surface.

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

About 90 minutes.

Four pictures are shown depicting segments of equal areas, of a comet's elliptical orbit around the sun. How would we find the length of time it takes for the comet to cover each segment?

Although Kepler wrote his laws specifically to describe the orbits of the planets around the Sun, they apply more generally. Kepler's second law tells us that as an object moves around its orbit, it sweeps out equal areas in equal times. Because all the areas shown here are equal, the time it takes the comet to travel each segment must also be the same.

A projectile is launched upward at an angle of 70° 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?

An angle of 20° because these angles sum to 90°.

Projectile

Any object that moves through the air or through space, acted on only by gravity, and air resistance, if any. Due to gravity, projectiles have curved paths.

The crew of a cargo plane wishes to drop a crate of supplies on a target below. To hit the target, when should the crew drop the crate? Ignore air resistance.

Before the plane is directly over the target. At the moment it is released, the crate shares the plane's horizontal velocity. In the absence of air resistance, the crate would remain directly below the plane as it fell.

A red ball rolls off an elevated platform with an initial horizontal velocity of 2 m/s. Just as it rolls off the platform, a blue ball is dropped from the platform from rest. Which ball hits the ground first (assume the ground is horizontal and flat)?

Both balls hit the ground at the same time. Both balls have the same initial velocity in the vertical direction and experience the same acceleration. This means they will fall at the same rate and hit the ground at the same time.

A t the instant a cannon fires a cannonball horizontally over a level range, another cannonball held at the side of the cannon is released and drops to the ground. Which ball, the one fired downrange or the one dropped from rest, strikes the ground first?

Both hit the ground at the same time.

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

By the law of energy conservation, the sum will always be constant.

ellipse

Closed curve of oval shape wherein the sum of the distances from any point on the curve to two internal focal points is a constant.

parabola

Curved path followed by a projectile acting under the influence of gravity only

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

Due to the force of gravity the velocity changes. On the way up it maintains the vertical initial speed until it gradually decreases zero (due to gravity pulling downward on the object). The object reaches a momentary zero, after which it increases in velocity in the downward direction as it falls to the ground.

How can a projectile "fall around the Earth"?

Earth is shaped in such a way that for every 8000 m tangential to the Earth's surface drops 5 m. Basically, every 8000 m there is 5 m drop. Projectiles that are falling around Earth are traveling at a speed that allows it orbit earth instead of fall into it.

True or false: Space vehicles orbit at altitudes in excess of 150 km to be above both gravity and Earth's atmosphere.

False. What satellites are above is the atmosphere and air resistance—not gravity! It's important to note that Earth's gravity extends throughout the universe in accord with the inverse-square law

True or false: Space vehicles orbit at altitudes in excess of 150 km to be above both gravity and Earth's atmosphere.

False. What satellites are above is the atmosphere and air resistance—not gravity! It's important to note that Earth's gravity extends throughout the universe in accord with the inverse-square law.

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

Gravitational force does not act horizontally. Gravity only affects the velocity of something vertical.

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

Gravity only affects the vertical component. The horizontal initial speed was not zero since the stone was thrown at an angle. No other net force acts in the horizontal direction therefore it remains constant throughout the flight.

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

Greatest nearest Earth; lowest furthest from Earth

A boy throws a ball from a tower 20 m down range. The vertical distance to the ground is 5 m. What is his pitching speed?

His pitching speed is the horizontal distance divided by time. Therefore v = d/t. Time is not stated, however, a 5 m drop indicates 1 second from the equation d = ½gt². Therefore, pitching speed is 20 m/s.

Predict how the vertical component of the velocity will change with time after the projectile is fired.

It first decreases to zero and then increases in the opposite direction.

Is the following explanation valid? Satellites remain in orbit instead of falling to Earth because they are beyond the main pull of Earth's gravity.

No, If any moving object were beyond the pull of gravity, it would move in a straight line and would not curve around Earth. Satellites remain in orbit because they are being pulled by gravity, not because they are beyond it. For the altitudes of most Earth satellites, Earth's gravitational field is only a few percent weaker than it is at Earth's surface.

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

No. Gravity always acts downward. At some point a projectile will reach it's maximum height, reach a momentary velocity of zero, and then begin falling back down with increased velocity.

Perigee

Point in an elliptical orbit closest to the focus about which orbiting takes place.

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

The farther the object is from Earth the weaker the force of gravity is. As an object orbits away from Earth its speed decreases. On the return to Earth the speed increases which propels it around the surface and back up again.

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

The force is at a right angle to the velocity. Gravity only pulls down ward. The satellite only moves in a direction perpendicular to the force of gravity that acts on it. It does not move in the direction of the force, otherwise speed would increase. There is no force to move an object forward or backward. Therefore, gravity is only pulling the satellite along its curves.

Kepler's second law

The line from the sun to any planet sweeps out equal areas of space in equal time intervals.

When no air resistance acts on a fast-moving baseball, its acceleration is

downward, g. Without air resistance, acceleration only acts on the vertical component.

Consider a planet with an elliptical orbit around the Sun. Its average distance from the Sun is __________.

half the distance along a line from the nearest to the farthest points in its orbit. The line that passes through the Sun and connects the nearest and farthest points in the orbit is called the major axis, and half this line is the semimajor axis — which is the planet's average distance from the Sun.

When you toss a projectile sideways, it curves as it falls. It will be an Earth satellite if the curve it makes

matches the curved surface of Earth. For an 8 km tangent, Earth curves downward 5m. Therefore a projectile traveling horizontally at 8 km/s will fall 5 m in that time and follow the curve of the Earth.

Kepler's third law states that for any planet orbiting the Sun, the orbital period squared is equal to the average orbital distance cubed. This implies that __________.

p² = a³. This implies a planet with a large average distance from the Sun has a longer orbital period than a planet with a smaller average distance from the Sun.

The velocity of a typical projectile can be represented by horizontal and vertical components. Assuming negligible air resistance, the horizontal component along the path of the projectile

remains the same. Without air resistance there is nothing to increase or decrease the horizontal component.

A cannonball is fired horizontally from the top of the highest mountain on Earth. If Earth didn't have an atmosphere, what initial speed is required such that the cannonball orbits around the Earth in a circle?

roughly 8 km/s. With this speed, it would take about 85 minutes to go around the Earth!

Without air drag the speed lost while going up equals____

the speed gained while coming down. Time going up, equals time coming down.

How does Earth's curvature relate to the speed needed for a projectile to orbit Earth?

the speed must be great enough so that the path of the projectile follows earth's curvature, instead of falling into it.

speed formula

v = d/t

Speed of a satellite in circular orbit formula

v = √gravity × mass / distance. G is the universal gravitational constant, M is the mass of Earth (or whatever body the satellite orbits), and d is the distance of the satellite from the center of Earth or other parent body.

The speed of a satellite in an elliptical orbit

varies. A satellite in an elliptical orbit cycles between receding from Earth and losing speed and approaching Earth and gaining speed.

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

Tycho Brahe gathered the information. Kepler discovered elliptical orbits. Newton combined everything and explained them.

What is the ratio of escape speed from Earth to circular orbital speed? Ignore air resistance.

1.41

Escape speed from Earth is any speed equal to or greater than __________.

11.2 km/s

Consider the video. (A ball is spring loaded into a cart and launched while in motion. The ball lands inside the cart). Suppose we replace the original launcher with one that fires the ball upward at twice the speed. We make no other changes. How far behind the cart will the ball land, compared to the distance in the original experiment?

4 times. The ball spends twice as long in the air. From d = ½gt², and 2² = 4. The distance itself will be 4 times as great. T is the only variable that we are changing.

For an Earth satellite in an elliptical orbit, list all the values that do change.

Speed, gravitational force, and distance from Earth

Kepler's First Law

The path of each planet around the Sun is an ellipse with the sun at one focus.

When a satellite travels at constant speed, the shape of its path is

a circle.

A rocket coasts in an elliptical orbit around Earth. To attain the greatest amount of KE for escape using 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 Fd=ΔKE 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.)

at the perigee.

A cannon, sitting on the ground in a flat horizontal field, fires a cannonball. For what angle between the orientation of the cannon and the ground will the cannonball land the furthest away from the cannon? Ignore air resistance

45 degrees. At this angle, the cannonball initially has the same horizontal and vertical velocities. If the angle were larger, it would stay in the air longer but wouldn't travel as far since its horizontal speed would be smaller. If the angle were smaller, it wouldn't stay in the air as long, so even though its horizontal speed would be faster, it wouldn't get as far before hitting the ground.

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

His third law. The square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun. This is for all planets. This means the ratio T²/r³ is the same for all planets. So if a planet's period is known, its average orbital radial distance is calculated.

Why does the force of gravity change the speed of a satellite when it is in an elliptical orbit but not when it is in a circular orbit?

In a circular orbit, the gravitational force is always perpendicular to the orbital path. With no component of gravitational force along the path, only the direction of motion changes—not the speed. In elliptical orbit, however, the satellite moves in directions that are not perpendicular to the force of gravity. Then components of force do exist along the path, which change the speed of the satellite. A component of force along (parallel to) the direction the satellite moves does work to change its KE.

Satellites in close circular orbit fall about 5 m during each second of orbit. Why doesn't this distance accumulate and send satellites crashing into Earth's surface?

In each second, the satellite falls about 5 m below the straight-line tangent it would have followed if there were no gravity. Earth's surface also curves 5 m beneath a straight-line 8-km tangent. The process of falling with the curvature of Earth continues from tangent line to tangent line, so the curved path of the satellite and the curve of Earth's surface "match" all the way around the planet. Satellites do, in fact, crash to Earth's surface from time to time when they encounter air resistance in the upper atmosphere that decreases their orbital speed.

A satellite is placed in orbit at the plane of Earth's equator. Why?

In order to appear motionless to an observer at a fixed location on Earth's surface, the satellite must rotate around Earth at the same rate as Earth rotates around its own axis and both the location and satellite must be in the same line between Earth's center and the satellite. a This can only occur above Earth's equator. Above any other location, the "ring" of satellite motion and the "ring" of motion of the location on Earth's surface would not be in the same plane.

When a projectile achieves escape speed from Earth, it does what?

It outruns the influence of Earth's gravity, but is never beyond it. All objects within the universe impart some gravitational force upon each other. That does not mean that the force is strong enough for the object to return.

Predict how the horizontal component of the velocity will change with time after the projectile is fired.

It stays constant.

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.

It will be slighter greater than 14 m/s. Find the resultant of the horizontal and vertical components.

Where does a satellite have the greatest KE? The greatest PE? The greatest total energy.

KE is maximum at the perigee, the closest point to Earth at the bottom of the track. PE is maximum at the apogee, the farthest point from Earth. The total energy is the same everywhere in orbit.

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

Kepler did not appreciate the concept of inertia. He believed that force on a moving body would be in the same direction as the body's motion. Newton realized that it is gravity that pulls objects toward each other.

Four pictures are shown depicting segments of equal areas, of a comet's elliptical orbit around the sun. How would we find and rank the distance the comet travels between each segment?

Kepler's second law tells us that the comet sweeps out equal areas in equal times. Because the area triangle is shorter and squatter for the segments nearer to the Sun, the distance must be greater for these segments in order for all the areas to be the same.

For an Earth satellite in circular orbit, list all the values that do not change.

Only speed, gravitational force, and distance from Earth

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

Orbital speed is less, the distance is greater, and the period is longer.

apogee

Point in an elliptical orbit farthest from the focus around which orbiting takes place.

Satellite

Projectile or smaller celestial body that orbits a larger celestial body

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

So air resistance or another friction force does not act upon the object. This way velocity is not slowed down and the object could not come crashing down.

period of satellite motion formula

T = 2π ×√distance³ / gravity × mass. G is the universal gravitational constant, M is the mass of Earth (or whatever body the satellite orbits), and d is the distance of the satellite from the center of Earth or other parent body.

engineering

Technology directed to the design, construction and maintenance of works, machinery, roads, railways, bridges, engines, all manner of vehicles from micro-carts to space stations, and to the generation, transmission, and use of electrical power. Some main divisions are aerospace, chemical, civil, communication, electrical, electronic, materials, mechanical, mining, and structural.

Two balls are thrown at the same time. One is thrown horizontally, the other is dropped. They both land at the same time. Which ball has the greatest speed at the moment of impact?

The ball thrown horizontally. The two balls have the same vertical velocity when they land, but the thrown ball has an additional horizontal velocity component. Since speed is defined as the magnitude of the resultant velocity vector, the thrown ball is moving faster when it lands.

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

The force of gravity is perpendicular to the motion in a circular orbit but not in an elliptical orbit.

A ball is thrown from the ground with a 45 degree angle between the initial velocity and the horizontal ground. When the ball reaches its highest point above the ground,

The horizontal component of the velocity and the acceleration of the ball are the same as right after the ball was thrown. The acceleration is always that of gravity. If it were zero, then the ball would move in a straight line from that location. There is no acceleration in the horizontal direction, so the horizontal speed doesn't change. Also, the acceleration is solely due to gravity, which doesn't change throughout the trajectory.

A red ball rolls off an elevated platform with an initial horizontal velocity. As it is falling down, how does the horizontal component of its velocity change (ignore air resistance)?

The horizontal component of the velocity does not change as the ball falls. Since the acceleration of the ball is solely due to gravity, which points directly downward, there is no acceleration in the horizontal direction.

Suppose Earth doubled in mass. How would Earth's orbital period around the Sun change?

The period would remain one year. Kepler's third law tells us that for a planet orbiting the Sun, the planet mass does not affect the orbital period

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?

The projectile will return at the same speed off 100 m/s.

What is an ellipse?

The set of points in which the combined distance from the two foci is the same. This is only true if the two foci are at the same location, which makes a special ellipse called a circle. This is why you can draw an ellipse using string, with both ends tacked to the two foci.

A bowling ball is in a circular orbit above the atmosphere of the Earth. Does the speed of the ball change throughout its orbit?

The speed of the ball is constant. The ball is never falling or rising away from the Earth, so its potential energy never changes. Thus its kinetic energy never changes.

Kepler's Third law

The square of the orbital period of a planet is directly proportional to the cub of the average distance of the planet from the Sun, T² ~ r³ or all planets.

A cannonball is fired horizontally from the top of the highest mountain on Earth with an initial velocity of 10 km/s (this speed is higher than the speed required to make it orbit in a circle, but less than the escape velocity of the cannonball). If Earth didn't have an atmosphere, what shape does the trajectory of the cannonball have?

The trajectory is an ellipse. The trajectory of any orbiting object is an ellipse

A baseball is batted at an angle into the air. Once airborne, and ignoring air drag, what is the ball's acceleration vertically? Horizontally?

The vertical acceleration is g (directed downward) because the force of gravity is vertical. The horizontal acceleration is zero because no horizontal force acts on the ball.

Consider a batted baseball that follows a parabolic path on a day when the Sun is directly overhead. How does the speed of the ball's shadow across the field compare with the ball's horizontal component of velocity?

They are the same!

Escape velocity

Velocity that a projectile, a space probe, etc, must reach to escape the gravitational influence of Earth or the celestial body to which it is attracted.

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?

Yes, escape speed can be less than 11.2 km/s if the speed is sustained long enough.

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.

You, consider the resultant.

horizontal distance formula

d = vt

Distance for free fall

d = ½gt²