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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

Using relativistic expressions, compare the momentum of two electrons, one moving at \(2.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\) and the other moving at \(2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\). What is the percentage difference between classical momentum values and these values?

Show that \(E^{2}-p^{2} c^{2}=E^{2}-p^{2} c^{2},\) that is, that \(E^{2}-p^{2} c^{2}\) is a Lorentz invariant. Hint: Look at derivation showing that the space-time interval is a Lorentz invariant.

Use light cones and world lines to help solve the following problem. Eddie and Martin are throwing water balloons very fast at a target. At \(t=-13 \mu s,\) the target is at \(x=0,\) Eddie is at \(x=-2 \mathrm{~km},\) and Martin is at \(x=5 \mathrm{~km},\) and all three remain in these positions for all time. The target is hit at \(t=0 .\) Who made the successful shot? Prove this using the light cone for the target. When the target is hit, it sends out a radio signal. When does Martin know the target has been hit? When does Eddie know the target has been hit? Use the world lines to show this. Before starting to draw your diagrams, consider: If your \(x\) position is measured in \(\mathrm{km}\) and you are plotting \(t\) versus \(x / c,\) what units must \(t\) be in, to the first significant figure?

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 \(0.650 c .\) This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of \(0.914 c\) to establish a relative speed of \(0.650 c\) with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

Consider a particle moving with a speed less than \(0.5 c\). If the speed of the particle is doubled, by what factor will the momentum increase? a) less than 2 b) equal to 2 c) greater than 2
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