Exam 26: The Special Theory of Relativity

Astronaut Hans leaves Earth in a spaceship traveling at a speed of 0.280c relative to the observer Brian at rest on Earth. Hans is holding a pencil at an angle of 30° with the direction of travel, as Seen by Hans. What angle does the pencil make with the direction of travel, as seen by Brian on Earth?

If a typical city consumes electrical energy at a rate of $2.0 \mathrm { GW }$ , how many kilograms of matter would have to be converted entirely into electrical energy in order to keep this city running for 14 weeks? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

Two spaceships are traveling through space at speeds of 0.600c and 0.900c, respectively, with respect to Earth. If they are headed directly toward each other, what is their approach speed (in terms of the speed of light) as measured by the captain of either ship?

A person with an initial mass of $70 \mathrm {~kg}$ climbs a stairway and thereby rises $6.0 \mathrm {~m}$ in elevation. By how much does his mass increase by virtue of his increased potential energy? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

A star is moving toward Earth at 90% the speed of light. The star emits light, which moves away from the star at the speed of light. What do Earth-based astronomers measure for the speed of this Light?

A particle is moving at $0.86 c$ . By what magnitude percentage is the Newtonian expression for momentum in error? (The percentage error is the difference between the erroneous and correct expressions, relative the correct one).

An object with a rest mass of $0.456 - \mathrm { kg }$ is moving at $1.20 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ . What is the magnitude of its momentum? $\left( c = 3.00 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

Two spaceships approach each other along a straight line at a constant relative velocity of 0.9880c as measured by the captain of one of the ships. An observer on Earth is able to measure the speed of only one of the ships as 0.9000c. From the point of view of the observer on Earth, what is the speed of the other ship in terms of the speed of light?

A particle physicist observes cosmic rays creating a new particle high in the atmosphere, and the speed of this particle is measured at $99.7 \%$ the speed of light. It is unstable, and is observed to decay in an average $37.0 \mu \mathrm { s }$ . If this particle were at rest in the laboratory, what would be its average lifetime?

A spaceship leaves Earth at a steady forward speed of 70% the speed of light. How fast relative to the ship must the spaceship launch a shuttlecraft in the forward direction so that it will be moving Away from Earth at 98.7% the speed of light?

A stretch of land is 12.5 km long on a map. (a) What would be its length to an observer moving by, parallel to the stretch, at 0.9500c? (b) Which is the proper length for this stretch of land? (c) How fast relative to Earth would the observer have to move for the land to appear 6.50 km long? Express your answer in terms of the speed of light.

One of the first atomic explosion released approximately $1.0 \times 10 ^ { 14 } \mathrm {~J}$ of energy. How much matter had to be changed into energy to produce this yield? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

A spaceship approaches Earth with a speed $0.50$ c. A passenger in the spaceship measures his heartbeat as 70 beats per minute. What is his heartbeat rate according to an observer that is at rest relative to Earth?

How fast (in terms of the speed of light) should a moving clock travel if it is to be observed by a stationary observer as running at one-half its normal rate?

A spaceship approaching an asteroid at a speed of 0.60c launches a scout rocket with speed 0.40c relative to the spaceship. At what speed is the scout rocket approaching the asteroid?

Suppose one twin takes a ride in a spaceship traveling at a very high speed to a distant star and then back again, while the other twin remains on Earth. Compared to the twin who remained on Earth, the astronaut twin is

When an object moves close to the speed of light relative to Earth, what happens to its dimensions compared to what they were before it began moving, as measured by an Earth-based observer?

Which of the following are basic postulates of special relativity? (There could be more than one correct choice.)

As an electron slows down from $0.9980 c$ to $0.5000 c$ , how much momentum does it lose? $( c$ $= 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } , m _ { \text {electron } } = 9.11 \times 10 ^ { - 31 } \mathrm {~kg}$ )

You are moving past the Earth at 0.990c and notice your heart beating 88 beats per minute. (a) If a doctor on Earth measures your heart rate as you zip past, what will the measured rate be? (b) If you had been moving 99.99% the speed of light, what rate would you have observed your heart beats?