Chapter 13: Newton’s Theory of Gravity

A rogue black hole with a mass $12$times the mass of the sun drifts into the solar system on a collision course with earth. How far is the black hole from the center of the earth when objects on the earth’s surface begin to lift into the air and “fall” up into the black hole? Give your answer as a multiple of the earth’s radius.

An object of mass $m$is dropped from height $h$above a planet of mass $M$and radius $R.$Find an expression for the object’s speed as it hits the ground.

A projectile is shot straight up from the earth’s surface at a speed of $10,000km/h$. How high does it go?

The unexplored planet Alpha Centauri III has a radius of $7.0\times {10}^{6}m$. A visiting astronaut drops a rock, from rest, into a $100-m$-deep crevasse. She records that it takes $6.0s$for the rock to reach the bottom. What is the mass of Alpha Centauri III?

An astronaut circling the earth at an altitude of $400km$is horrified to discover that a cloud of space debris is moving in the exact same orbit as his spacecraft, but in the opposite direction. The astronaut detects the debris when it is $25km$away. How much time does he have to fire his rockets and change orbits?

Suppose that on earth you can jump straight up a distance of $45cm.$Asteroids are made of material with mass density $2800kg/m^3.$What is the maximum diameter of a spherical asteroid from which you could escape by jumping?

A rogue band of colonists on the moon declares war and prepares to use a catapult to launch large boulders at the earth. Assume that the boulders are launched from the point on the moon nearest the earth. For this problem you can ignore the rotation of

the two bodies and the orbiting of the moon.

a. What is the minimum speed with which a boulder must be launched to reach the earth?

Hint: The minimum speed is not the escape speed. You need to analyze a three-body system.

b. Ignoring air resistance, what is the impact speed on earth of a boulder launched at this minimum speed?

Two spherical asteroids have the same radius $R$. Asteroid$1$has mass $M$and asteroid $2$has mass $2M.$The two asteroids are released from rest with distance $10R$between their centers. What is the speed of each asteroid just before they collide?

Hint: You will need to use two conservation laws.

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet’s surface. How far above the surface are they orbiting? Your answer will be a multiple of $R$.

A space station astronaut is working outside the station as it orbits the earth. If he drops a hammer, will it fall to earth? Explain why or why not.