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Chapter 35: Problem 64

Rocket A passes Earth at a speed of \(0.75 c\). At the same time, rocket B passes Earth moving \(0.95 c\) relative to Earth in the same direction. How fast is B moving relative to A when it passes \(A ?\)

Short Answer

Expert verified
Answer: The relative speed of Rocket B with respect to Rocket A is approximately 0.9926c.

Step by step solution

01

1. Understand the relativistic velocity addition formula

The relativistic velocity addition formula is given by: $$ V = \frac{u+v}{1+\frac{uv}{c^2}} $$ where \(V\) is the relative speed of the two objects, \(u\) is one object's speed relative to a reference frame (in this case, Earth), \(v\) is the other object's speed relative to the same reference frame, and \(c\) is the speed of light.
02

2. Substitute the given values into the formula

In our problem, the speed of Rocket A relative to Earth is \(u = 0.75c\), and the speed of Rocket B relative to Earth is \(v = 0.95c\). We'll now substitute these values into the relativistic velocity addition formula: $$ V = \frac{(0.75c) + (0.95c)}{1 + \frac{(0.75c)(0.95c)}{c^2}} $$
03

3. Simplify the expression

We can simplify the expression by factoring out \(c\) in the numerator and canceling out with the \(c^2\) in the denominator: $$ V = \frac{c(0.75 + 0.95)}{1 + \frac{0.75 \cdot 0.95}{1}} $$
04

4. Calculate the relative speed

Now, we can compute the relative speed by solving for V: $$ V = \frac{c(1.7)}{1 + 0.7125} $$ $$ V = \frac{1.7c}{1.7125} $$ $$ V \approx 0.9926c $$ So, Rocket B is moving at a speed of approximately \(0.9926c\) relative to Rocket A when it passes Rocket A.

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Most popular questions from this chapter

A gravitational lens should produce a halo effect and not arcs. Given that the light travels not only to the right and left of the intervening massive object but also to the top and bottom, why do we typically see only arcs?

The explosive yield of the atomic bomb dropped on Hiroshima near the end of World War II was approximately 15.0 kilotons of TNT. One kiloton is about \(4.18 \cdot 10^{12} \mathrm{~J}\) of energy. Find the amount of mass that was converted into energy in this bomb.

The hot filament of the electron gun in a cathode ray tube releases electrons with nearly zero kinetic energy. The electrons are next accelerated under a potential difference of \(5.00 \mathrm{kV}\), before being steered toward the phosphor on the screen of the tube. a) Calculate the kinetic energy acquired by the electron under this accelerating potential difference. b) Is the electron moving at relativistic speed? c) What is the electron's total energy and momentum? (Give both values, relativistic and nonrelativistic, for both quantities.)

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.65 c\). a) Calculate the total distance Alice traveled during the trip, as measured by Alice. b) With the aforementioned total distance, calculate the total time duration for the trip, as measured by Alice.

Find the speed of light in feet per nanosecond, to three significant figures.
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