Concept explainers
Two powerless rockets are on a collision course. The rockets are moving with speeds of 0.800c and 0.600c and are initially 2.52 × 1012 m apart as measured by Liz, an Earth observer, as shown in Figure P1.34. Both rockets are 50.0 m in length as measured by Liz. (a) What are their respective proper lengths? (b) What is the length of each rocket as measured by an observer in the other rocket? (c) According to Liz, how long before the rockets collide? (d) According to rocket 1, how long before they collide? (e) According to rocket 2, how long before they collide? (f) If both rocket crews are capable of total evacuation within 90 min (their own time), will there be any casualties?
Figure P1.34
(a)
The proper length of the two rockets.
Answer to Problem 34P
The proper length of rocket 1 is
Explanation of Solution
Write the formula for length contraction
Here,
Rearrange equation (I) and solve for
Substitute
Thus, the proper length of rocket 1 is
Substitute
Thus, the proper length of rocket 2 is
Conclusion:
Thus, the proper length of rocket 1 is
(b)
The length of each rocket as measured by an observer in the other rocket.
Answer to Problem 34P
The contracted length of rocket 2 measured by an observer in rocket 1 is
Explanation of Solution
Write the expression of Lorentz velocity transformation
Here,
Substitute
Thus, the velocity of rocket 2 with respect to rocket 1 is
Substitute
The contracted length of rocket 2 measured by an observer in rocket 1 is
Substitute
The contracted length of rocket 1 measured by an observer in rocket 2 is
Conclusion:
Thus, the contracted length of rocket 2 measured by an observer in rocket 1 is
(c)
The time takes for the rockets to collide according to Liz.
Answer to Problem 34P
The time takes for the rockets to collide according to Liz is
Explanation of Solution
The velocity of the rocket 1 is
Write the formula to find the time with respect to distance
Here,
Substitute
Conclusion:
Thus, the time takes for the rockets to collide according to Liz is
(d)
The time takes for the rockets to collide according to rocket 1.
Answer to Problem 34P
The time takes for the rockets to collide according to rocket 1 is
Explanation of Solution
Substitute
Substitute
Conclusion:
Thus, the time takes for the rockets to collide according to rocket 1 is
(e)
The time takes for the rockets to collide according to rocket 2.
Answer to Problem 34P
The time takes for the rockets to collide according to rocket 2 is
Explanation of Solution
Substitute
Substitute
Conclusion:
Thus, the time takes for the rockets to collide according to rocket 2 is
(f)
Whether there will be any casualties in the rockets during the collision.
Answer to Problem 34P
There will be no casualties in rocket 2 during collision while there will be some casualties left in rocket 1 during collision.
Explanation of Solution
Both the rocket crews are capable of total evacuation with
From part (d), the time takes for the rockets to collide according to rocket 1 is
From part (e), the time takes for the rockets to collide according to rocket 2 is
Conclusion:
Therefore, there will be no casualties in rocket 2 during collision while there will be some casualties left in rocket 1 during collision.
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Chapter 1 Solutions
EBK MODERN PHYSICS
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