Scientific Inquiry and Analysis: Physics (Gravity and Projectiles)

A payload into orbit requires control over...

- direction of rocket - speed of rocket

Projectile curves as a result of these 2 components:

1. constant motion horizontally - it rolls of its own inertia and covers equal distances in equal amounts of time 2. accelerated motion vertically - just like free fall - the faster an object falls, the greater the distance covered in each second

Intensity formula

1/d^2

Escape speed is ______ kilometers per second from Earth's surface

11.2

Calculate the force of gravity on the same 1-kg mass if it were 6.4 x 10^6 m above Earth's surface (that is, if it were two Earth radii from Earth's center

2.5 N

As the ball leaves the girl's hand, one second later it will have fallen...

5 meters below the dashed line. Whatever the speed, the ball will fall a vertical distance of 5 meters below the dashed line.

When an elevator accelerates upward, your weight reading on a scale is...

greater. The support force pressing on you is greater, so you weigh more.

To become an Earth satellite, the projectile's ______________ velocity must be great enough for its trajectory to match Earth's curvature (8 km/s).

horizontal

To become an Earth satellite, the projectile's ________________ velocity must be great enough for its trajectory to match Earth's _____________.

horizontal, curvature

Circular Satellite Orbits (direction of rocket)

initially, rocket is fired vertically, then tipped once above the atmosphere, the rocket is aimed horizontally

Weightlessness

is a condition wherein a support force is lacking—free fall,

Projectile

is any object that moves through the air or through space under the influence of gravity.

Weight

is the force exerted against a supporting floor or weighing scale.

When an elevator accelerates downward, your weight reading is...

less The support force pressing on you is less, so you weigh less.

The force of gravity between two planets depends on their...

masses and distance apart. The equation for gravitational force, cites only masses and distances as variables. Rotation and atmospheres are irrelevant.

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

In a circular orbit, the speed of a satellite is ____________________ by gravity, only direction changes.

not changed

When a projectile achieves escape speed from Earth, it...

outruns the influence of Earth's gravity but is never beyond it.

Tangential velocity

parallel to the earth's surface

Circular Satellite Orbits (speed of rocket)

payload is given a final thrust to orbital speed of 8 km/s to fall around Earth and become an Earth satellite

If the masses of two planets are each somehow doubled, the force of gravity between them...

quadruples. Note that both masses double, so double × double = quadruple.

The further a satellite is from earth, the _____________ its speed, the ___________ its path, and the __________ its period

slower, longer, longer

Elliptical orbit

speed of satellite varies -initially, if speed is greater than needed for circular orbit, satellite overshoots a circular path and moves away from Earth -satellite loses speed and then regains it as it falls back toward Earth -it rejoins its original path with the same speed it had initially -procedure is repeated

Escape speed

the initial speed that an object must reach to escape gravitational influence of Earth 11.2 kilometers per second from Earth's surface

(Projectile Motion) Without air resistance...

the speed lost going up is the same as the speed gained while coming down.

The Effect of Air Resistance on Projectiles (without air resistance)

the speed lost going up is the same as the speed gained while coming down.

Parabola

the trajectory of a projectile that accelerates only in the vertical direction while moving at a constant horizontal velocity

A ball tossed at an angle of 30° with the horizontal will go as far downrange as one tossed at the same speed at an angle of...

60°. Same initial-speed projectiles have the same range when their launching angles add up to 90°. Why this is true involves a bit of trigonometry—which in the interest of time, we'll not pursue here.

escape velocity equation

v = square root of (2GM/R)

In an elliptical orbit, the speed of satellite ___________

varies

The speed of a satellite in an elliptical orbit...

varies.

When an elevator cable breaks your weight is...

zero weight

The Universal Gravitational Constant, G

G is the proportionality constant in Newton's law of gravitation. G has the same magnitude as the gravitational force between two 1-kg masses that are 1 meter apart: 6.67 × 10-11 N. So G = 6.67 × 10-11 N ⋅ m2/kg2. F = 6.67 × 10-11 N ⋅ m2/kg2(m1 × m2)/d2

When an elevator goes up your weight is...

Greater than normal weight

When an elevator goes down your weight is...

Less than normal weight

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

Speed doesn't change when there is no component of gravitational force in the direction of its motion

What happens to a satellite close to Earth's surface if it is given a speed exceeding 11.2 km/s?

Such a speed is called the escape speed because the satellite escapes the influence of Earth

Horizontal Projectiles (Examples)

The ball's horizontal component of velocity doesn't change as the falling ball moves forward (no force is acting horizontally) The ball's vertical component of velocity does change due to gravity. The vertical distances become further apart with time

Would the springs inside a bathroom scale be more compressed or less compressed if you weighed yourself in an elevator that moved upward at constant velocity? Downward at constant velocity?

The compression does not change at constant velocity

Upon which is the gravitational force greater: a 1-kg piece of iron or a 1-kg piece of glass? Defend your answer

The force of gravity is the same on each because the masses are the same, as Newton's equation for gravitational force verifies

Satellite

any projectile moving fast enough to fall continually around the Earth (without falling into it).

The greater the ____________ from Earth, the less the gravitational force on an object. No matter how great the distance, gravity approaches, but never reaches, ________.

distance zero

If the mass of one planet is somehow doubled, the force of gravity between it and a neighboring planet would...

double. Use the equation for Newton's Law of Universal Gravitation. Note that if one mass doubles, the force between them doubles.

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

downward, g.

Newton's most celebrated synthesis is of...

earthly and heavenly laws. This synthesis provided hope that other natural phenomena followed universal laws, and ushered in the Age of Enlightenment.

Escape velocity

escape speed when direction is involved

The greater m1 and m2 ⇒ the greater the _________ of attraction between them. The greater the ______________ of separation d, the weaker is the force of attraction—___________ as the inverse square of the distance between their centers.

force distance, weaker

The greater the masses of interacting objects, the __________________ the gravitational force between them

greater

The planet Prinzo is about 200 times as massive as Earth, yet on its surface you would weigh only about 2 times as much. This is because planet Prinzo's radius is ________________ than earth's radius

greater

When the elevator cable breaks, the elevator falls freely so your weight reading is...

zero. There is still a downward gravitational force acting on you, but gravity is not felt as weight because there is no support force. So your weight is zero.

The universal gravitational constant, G, which links force to mass and distance, is similar to the familiar constant...

π. Just as π relates the circumference of a circle to its diameter, G relates force to mass and distance.

Calculate the force of gravity between a newborn baby of mass 3.0 kg and the obstetrician of mass 100 kg, who is 0.5 m from the baby. Which exerts more gravitational force on the baby: Mars or the obstetrician? By how much?

8.0 x 10^-8 The obstetrician exerts twice the gravitational force on the baby

The Earth's curvature drops a vertical distance of 5 meters for each 8000 m tangent to the surface. So to orbit Earth, a projectile must travel ______ m in the time it takes to fall 5 m (18,000 miles/hour).

8000

For equal launching speeds, the same range is obtained from two different projection angles—a pair that add up to ______ degrees

90

If Superman tosses a ball 8 km/s horizontally from the top of a mountain (just above air resistance): what could happen to the ball?

90 minutes later - he can turn around and catch it Tossed slightly faster - it takes an elliptical orbit and takes slightly longer to return Tossed at more than 11.2 km/s - it escapes earth Tossed at more than 42.5 km/s - it escapes the solar system

Why is it important that a satellite remain above Earth's atmosphere

A satellite must remain above Earth's atmosphere because air resistance would not only slow it down but also incinerate it at its high speed. A satellite must not contend with either of these

Curved path of projectile (parabola) Example

A stone thrown horizontally curves downward due to gravity.

Newton's Law of Universal Gravitation

Every body in the universe attracts every other body with a mutually attracting force. For two bodies, this force is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them,

Circular Satellite Orbits (positioning) Example

Example: Space shuttles are launched to altitudes of 150 km or more, to be above air drag. (Are they above Earth's gravitational)

Newton's Law of Universal Gravitation equation

F = G(m1*m2/r2) G = 6.67 x 10^-11 N

Circular Satellite Orbits (continued)

For a satellite close to earth, the period (time to make a complete orbit around the earth) is 90 minutes The further a satellite is from earth, the slower its speed, the longer its path, and the longer its period The moon has a period of 27.3 days!

Projectile Altitude and Range

For equal launching speeds, the same range is obtained from two different projection angles—a pair that add up to 90°.

If a flight mechanic drops a box of tools from a high-flying jumbo jet, the box crashes to Earth. If an astronaut in an orbiting space vehicle drops a box of tools, does it crash to Earth also?

If a box of anything is "dropped" from an orbiting space vehicle, it has the same tangential speed as the vehicle and remains in orbit. If a box is dropped from a high-flying jumbo jet, it too has the tangential speed of the jet. But this speed is insufficient for the box to fall around and around Earth. Instead it soon falls to Earth's surface

Circular Satellite Orbits (Examples)

If a cannonball could be launched at 8 km/s, it would follow earth's curvature and glide in a circular path around earth If it were fired at a slower speed, it would crash back into earth If it were fired at a faster speed, it would overshoot a circular orbit - out to space In a circular orbit, the speed of a satellite is not changed by gravity, only direction changes

What if Projectiles are Launched at an Angle

If a cannonball is shot at an upward angle, it should follow a straight line path (no gravity) But with gravity, the cannonball falls until it finally reaches the ground So, where does the cannonball end up? D = ½ g t^2

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

In an elliptical orbit there is a component of force in the direction of motion

The Fact of the Falling Moon (Ancient Beliefs)

In ancient times, it was believed that stars, planets, and the Moon moved in divine circles, free from forces of Earth. Newton recognized that some force must be acting on the planets, otherwise their paths would be straight lines. Newton also realized that any force on a planet would be directed toward a fixed, central point - the sun. Newtonian Synthesis - the union of terrestrial and cosmic laws.

A stone is thrown upward at an angle. Neglecting air resistance, what happens to the vertical component of its velocity along its trajectory?

Neglecting air resistance, the vertical component of velocity decreases as the stone rises and increases as it descends, the same as with any freely falling object

Does the speed of a falling object in the absence of air resistance depend on its mass? Does the speed of a satellite in orbit depend on its mass?

Neither the speed of a falling object (without air resistance) nor the speed of a satellite in orbit depends on its mass. In both cases, a greater mass (greater inertia) is balanced by correspondingly greater gravitational force, so the acceleration remains the same

What did Newton discover about gravity?

Newton discovered that gravity is universal

The Legend of the Falling Apple

Newton was not the first to discover gravity. Newton discovered that gravity is universal. Legend—Newton, sitting under an apple tree, realized that the force between Earth and the apple is the same as that between moons and planets and everything else.

Example of weightlessness

No support force. An astronaut is weightless because he or she is not supported by anything. The body responds as if gravity forces were absent, and this gives the sensation of weightlessness.

When an elevator is still your weight is..

Normal weight

The Universal Gravitational Constant, G (continued)

Once the value of G was known, the mass of earth could be calculated: F = G (m1 x m2 / d2) M1 (the mass of the earth) was determined to be: 6 x 10^24 kg

First probe to escape the solar system is _________________, launched from Earth in 1972.

Pioneer 10

Projectile Altitude and Range (Example)

Same range occurs for a 75° launch and a 15° launch of the same initial

What is the magnitude of the gravitational force between Earth and a 1-kg body at its surface?

The gravitational force is 9.8 N

The hang time of a basketball player who jumps a vertical distance of 2 ft (0.6 m) is about 0.6 s. What is the hang time if the player reaches the same height while jumping 4 ft (1.2 m) horizontally?

The hang time is the same, in accord with the answer to the preceding exercise. Hang time is related to the vertical height attained in a jumpy, not to the horizontal distance moved across a level floor.

Inverse square law

The intensity of an effect from a localized source spreads uniformly throughout the surrounding space and weakens with the inverse square of the distance Gravity follows an inverse-square law, as do the effects of electric, light, sound, and radiation phenomena

The Inverse-Square Law: Gravity and Distance

The inverse square law holds for gravity and for all phenomena in which the effect from a localized source spreads uniformly throughout the surrounding space We only sense gravitation when large masses are involved. The force of attraction between you and the earth is your weight. Your weight depends on both your mass and your distance from the center of the earth. A child that weighs 300N at sea level, weighs only 299N on Mount Everest - and much less the further away from earth he gets - but he will never weigh zero.

Ellipse

The oval path followed by a satellite. The closed path taken by a point that moves in such a way that the sum of its distances from two fixed points (called foci) is constant. The closer the foci to each other, the more circular it gets.

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

The projectile returns at the same speed of 100 m/s

What two forces act on you while you are in a moving elevator? When are these forces of equal magnitude, and when are they not?

The two forces are the normal force and mg, which are equal when the elevator doesn't accelerate and unequal when the elevator accelerates

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

There are no forces acting horizontally (neglecting air resistance) so there is no horizontal acceleration; hence the horizontal component of velocity doesn't change. Gravitation acts vertically, which is why the vertical component of velocity changes

An astronaut lands on a planet that has the same mass as Earth but half the diameter. How does the astronaut's weight differ from that on Earth?

Use the equation for gravitation to guide your thinking: Half the diameter is half the radius, which corresponds to 4 times as much weight as on Earth

The Fact of the Falling Moon (Current Beliefs)

We now know that the Moon falls around Earth in the sense that it falls beneath the straight line it would follow if no force acted on it. The Moon maintains a tangential velocity, which ensures a nearly circular motion around and around Earth rather than into it. This path is similar to the paths of planets around the Sun.

Weight and Weightlessness (Examples)

When you step on a scale, you are compressing a spring inside the scale Once the elastic force of the deformed spring balances the gravitational attraction between the earth and you, the pointer stops at your weight If you stand on a scale in an accelerating elevator you will find variations in your weight: -Elevator accelerates upward - springs in scale are more compressed - you weigh more -Elevator accelerates downward - springs in scale are less compressed - you weigh less -If elevator cable breaks and elevator falls freely - you would be weightless!

If a satellite were beyond Earth's gravity, what path would it follow?

With no gravity, the path of a moving object would be a straight line, in accord with the law of inertia

To begin your wingsuit flight, you step off the edge of a high cliff. Why are you then momentarily weightless? At that point, is gravity acting on you?

You are weightless because there is no support force. The force of gravity is nevertheless acting on you and will be the only force on you until air resistance builds up

Where do you weight more: at sea level or no top of one of the peaks of the Rocky Mountains?

You're closer to Earth's center at sea level, so you weigh more there than on any mountain peak.

When is your weight equal to mg?

Your weight is mg when you are firmly supported and in equilibrium

If Earth somehow expanded to have a larger radius, with no change in mass, how would your weight be affected? How would it be affected if Earth instead shrunk?

Your weight would decrease if Earth expanded with no change in its mass and would increase if Earth contracted with no change in its mass. Your mass and Earth's mass don't change, but the distance between you and Earth's center does change. Force is proportional to the inverse square of this distance

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

a circle.

Projectile Motion

a combination of a horizontal component and a vertical component

If you weigh yourself in an elevator, you'll weigh more when the elevator...

accelerates upward. The support provided by the floor of an elevator is the same whether the elevator is at rest or moving at constant velocity. Only accelerated motion affects weight.

Circular Satellite Orbits (positioning)

beyond Earth's atmosphere, where air resistance is almost totally absent

(Projectile Motion) With air resistance...

both range and altitude are decreased

The Effect of Air Resistance on Projectiles (with air resistance)

both range and altitude are decreased

2 components of projectile motion - Horizontal ________________________ - Vertical ______________________

constant velocity acceleration due to gravity

If the mass of Earth somehow decreased with no change in radius, your weight would ______________________

decrease