Gravity and Escape Velocity Tutorial

Gravity is the force that gives most of the large objects in the Universe their structure. It holds the Earth together and in its orbit around the Sun. It holds the Sun together and in its orbit around the Milky Way. It even holds the Universe itself together.

Gravity also causes many astronomical phenomena such as tides. As perhaps the most important force for large objects, we need to understand its basic properties.

Newton discovered the mathematical expression for the force of gravity and showed it has the following form.

Force of gravity = constant x Mm/r2, Where M and m are the masses and r is their separation. The constant is called the Universal gravitational constant and its value depends on the units used.

If the masses are in kilograms (kg) and the distance is measured in meters, and the force in metric units called newtons, then G =6.67x10-11 newton-m2-kg-2.

The Newton is a unit of force and is defined so that a force of 1 newton produces an acceleration of 1 meter per second2 on a mass of 1 kilogram.

A little fiddling then shows that the Newton can also be expressed in more fundamental units so that the units of the newton = kilogram-meter-sec-2=kg-m-sec-2.

Thus the units of G can be expressed as meters3-kg-1-sec-2.

The critical ideas to understand are simple.

Notice that doubling the distance doesn't reduce the force by a factor of 2. It reduces it by the square of the distance, or in this case, a factor of 4.

For example, suppose the Earth were 3 times farther from the Sun than it is now. How would the gravitational force between them change? Answer: It would be 32 = 9 times weaker.

1. Suppose the Moon were five times closer to the Earth than it is now. How would the gravitational force between them change?
A)..It would be 5 times weaker.
B).. It would be five times stronger.
C).. It would be 10 times stronger.
D).. It would be 25 times stronger.
E).. It would be 125 times stronger.

2. Suppose two stars are in orbit around each other. If one star explodes and blows off half its mass, what will happen to the gravitational force between the pair?
A)...It will get twice as strong.
B)... It will get four times as strong.
C)...It will be only half as strong.
D)...It will be only one-quarter as strong.
E)... It will not change.

The force of gravity affects many aspects of astronomical objects. In particular, in governs the speed that something most travel in order to escape the object's gravity. That speed is called the escape velocity .

The formula for the escape velocity from a spherical object like a moon, planet, or star, is

V = (2GM/R)

where G is the gravitational constant, M is the object's mass, and R is its radius.

Note that for a given mass, as R gets smaller, V will get larger.

Notice also that the escape velocity formula looks a lot like the law of gravity, as you might expect. After all, a greater gravitational force of attraction will make it harder to escape an object and thus you must move faster to break free.

Suppose we have two planets A and B. They are of the same radius but planet A is more massive than planet B.

3. Which has the larger escape velocity?
A)... A
B)... B
C)... The mass doesn't affect the escape velocity.

4. Suppose in the above problem A's mass is 9 times B's. How does the escape velocity from A compare to B?
A)... It is 9 times smaller.
B)... It is 9 times larger.
C)... It is 3 times smaller.
D)... It is 3 times larger.
E)... They are the same.

Now let's actually calculate the escape velocity, V, for an object, for example, the Sun. The Sun's mass is about 2x1030 kg and its radius is about 7x108 meters. Given that G = 6.67x10-11, what is V? Note that when you work this problem, your answer will be in meters/sec.

Inserting the above numbers in the formula we find V = (2GM/R) = (2x6.67x10-11 x2x1030/7x108) = (2x6.67x2x10-11+30/7x108) = (4x1011) = 6.3x105 meters/sec.

This answer may be easier to interpret if we express it in kilometers/sec, rather than meters/sec. To make that conversion, recall that a kilometer is 1000 (=103)meters. Thus, 6.3x105 meters/sec = 6.3x105 meters/sec/103meters/km = 6.3x102 km/sec = 630 km/sec.

5.Repeat the above calculation to find the escape velocity from the Earth. The Earth's mass is 5.97x1024 kg and its radius is 6.38x106 meters. Make the same conversion from m/sec to km/sec at the end.
A)... 125 km/sec.
B)... 11.2 km/sec
C)...1.12 km/sec
D)... 12,500 km/sec.
E)... 1,120 km/sec.

Scientists often find it convenient to scale calculations, such as the one we just did so that the math need not be redone if we want to apply the calculations to another planet or object. For example, suppose we had a planet the same radius as the Earth but 25 times more massive. Because the escape velocity depends on the square root of the mass, to find the escape velocity for this more massive planet we simply multiply the Earth's escape velocity by the square root of 25 (=5) to find that its escape velocity is about 56 km/sec.

Similarly, if the planet has the same mass but a different radius, we can again scale the escape velocity. This time we divide by the square root of the factor by which the radius is larger.

If both the mass and radius differ from Earth's (as of course they generally will), apply both scaling factors. For example, Uranus is about 15 times more massive than Earth and its radius is about 4 times Earth's. What is its escape velocity?

The 15 times bigger mass means V is 15 = 3.87 times larger. The 4 times larger radius means that V is 4 = 2 times smaller. Combining the scaling factors gives us that V is 3.87/2 = 1.94 times larger or V = 1.94x11.2 km/sec = 21.7 km/sec.

6. What is the escape velocity from Jupiter given that its mass is 300 times that of the Earth's and its radius is 10 times larger. HintUse the scaling principles above.
A)... 6.13 km/sec.
B)... 333.2 km/sec
C)...1.12 km/sec
D)... 61.3 km/sec.
E)... 33.3 km/sec.

Although gravity holds most astronomical objects together, it can also tear objects apart. For example, if you could jump into a black hole of about 1 solar mass, you would be stretched and torn into atoms by its gravity.

A less dramatic example of gravity's stretching power is the tides. Ocean tides on Earth arise from the Moon's gravitational pull on the Earth (with some help from the Sun). A full discussion of the tides will come later. Here we will simply look at the general case of gravity stretching an object.

Suppose we put two small masses, A and B in line with a third larger one, C (see Figure). Three spheres For simplicity, let's ignore the gravitational attraction of the small objects and suppose that none of the three is moving.

7. What will happen to the smaller ones?
A)... They will remain where they are.
B)... They will feel a gravitational force from C and move toward it.
C)...They will feel a gravitational force from C but their inertia will keep them from moving.

8. Which of the two masses A or B feels the greater force?
A)... A.
B)... B.
C).. They both feel the same force.

9. Given that both A and B will move toward C, which will reach C sooner?
A)... A.
B)... B.

10. Why?
A)... B is closer to C and thus feels a stronger gravitational force than A.
B)... B is closer to C and thus shields A from C's gravity.

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