Learning Goal: To understand length contraction and time dilation. An inertial f

Question

Learning Goal: To understand length contraction and time dilation.

An inertial frame of reference is one in which Newton's laws hold. Any frame of reference that moves at a constant speed relative to an inertial frame of reference is also an inertial frame. The proper length l_0 of an object is defined to be the length of the object as measured in the object's rest frame. If the length of the object is measured in any other inertial frame, moving with speed u relative to the object's rest frame (in a direction parallel to l_0), the resulting length l is given by the length contraction equation

l = l_0sqrt{1-rac{u^2}{c^2}} ,

where c is the speed of light. Similarly, if two events occur at the same spatial point in a particular reference frame, and an observer at rest in this frame measures the time interval between these two events, the time interval so measured is defined to be the proper time Delta t_0. When the time interval is measured in any other inertial frame, again moving with speed u relative to the first frame, the resulting time interval Delta t is given by the time dilation equation

Delta t = rac{Delta t_0}{sqrt{1-rac{u^2}{c^2}}} .

a. Suppose that you measure the length of a spaceship, at rest relative to you, to be 400 m m. How long will you measure it to be if it flies past you at a speed of u = 0.75c?
Express the length l in meters to three significant figures.

b. The spaceship from Part A has a large clock attached to its side. This clock ran at the same rate as your watch when you were in the same reference frame. How much time Delta t will pass on your watch as 80 m s passes on the clock attached to the ship?
Express your answer in seconds to three significant figures.

c. Two spaceships, named A and B, are flying toward each other with relative speed 0.800c. If the captain of ship A fires a missile, counts 10.0 m s on his watch, and then fires a second missile, how much time Delta t will the captain of ship B measure to have passed between the firing of the two missiles?
Express your answer in seconds to three significant figures.

d. The captain of ship B knows that ship A uses 2-m-long missiles. She measures the length of the first missile, once it has finished accelerating, and finds it to be only 0.872 m m long. What is the speed u of the missile, relative to ship B?
Express your answer in meters per second to three significant figures. Use c = 3 imes 10^8 ; m m/s.

Please answer fully... thank you

Explanation / Answer

a.) 264.6m

c.) 16.7s

tB = 10*1/[1 -0.8*0.8] = 16.7 s

d.) l = l*[1 - u*u/c*c] = 2*[1 -u*u/c*c] = 0.872

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