Vectors and Energy

1.

The rectangular box shown below is 10 cm high (in the z direction), 32 cm wide (in the y direction) and 20 cm deep (in the x direction).

(a)

What is the angle CAB (in degrees)?

(b)

What is the shortest distance from point F to the line AB?

(c)

Give the coordinates [x,y,z] of the point D where the line FD meets the line AB at right angles.

2.

A frictionless roller coaster is built on a moon of another planet in some other solar system, as illustrated.

The moon has a mass $M = 8 \times 10^{-3} M_E$ and a radius R = 0.2 R_E.

A 1000 kg car starts from rest at position A 1 km above the moon's surface and slides down the roller coaster to position C on the moon's surface. What is its speed at position C?

3.

A rocket leaving the Moon, on a radial path, has just reached a speed of 2.0 km/s at an altitude of 1000 km above the surface, when a wrench falls off it. Will the wrench reach the Moon's surface? Justify your answer and evaluate either (a) the impact speed, if it hits, or (b) its speed as it reaches an altitude of 100,000 km above the surface, if it escapes.

4.

Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a horizontal table. The target box is 2.2 m horizontally from the edge of the table (see Figure). Bobby compresses the spring 1.10 cm and shoots the marble with it, but the marble falls 27 cm short. How far should Rhoda compress the spring to score a hit?

5.

For a light mass m in a circular orbit about a much heavier mass M, prove that the total energy  E = K + U  is half the gravitational potential energy  U,  where we adopt the usual convention $U \to 0$ as $r \to \infty$. (K is the kinetic energy.)

6.

Captain Kirk places the (old) starship Enterprise in a circular orbit about a 1-Solar-mass neutron star (radius 10 km) at a safe (???) distance of 12,000 km from its centre.

(a)

What is the centripetal acceleration of the centre of gravity of the Enterprise?

(b)

What is the period of its orbit?

(c)

If Kirk's head is 1.6 m further from the star than his feet, what is the ``tidal'' difference between the gravitational accelerations of his head and feet?