Question

Good morning. Today I will also advance some physics, the following exercises sent me but in these three exercises of the workshop I got confused by the reference system, I did not get to raise the equation. I thank you in advance for your help :)
First exercise:
According to the driver of a car, whose speed is 0.64c, the last lap of the circuit was done in only 18s. How long did it take to make that lap according to the public who attended the event? Make an outline of the previous situation.
Second exercise:
An observer passes by a certain place with a speed, along the horizontal axis, of 0.86c, and appreciates that the boom (or arm) of a crane has a length of 5.8m and forms an angle of 60° with respect to the Earth. According to crane operator:
a) How long is the boom?
b) At what angle did you lift it?
c) Make a diagram of the previous situations.
Third exercise:
The pilot of a rocket that moves at a speed of 0.77c with respect to the Earth observes a
second rocket approaching in the opposite direction with a velocity of 0.45c. What velocity will an observer on Earth measure for the second rocket? Make an outline of the above situation.

Answer 1

a) let's approach the first problem by Lorentz transformation

we know,

since t = 18 s

v= 0.64 c

x = vt = 18 s x 0.64 c

= 11.52 c

thus subs.

= 10.6272/ 0.6

= 17.712

thus for the people on the stand, will see the time to complete be 17.712 s

b) for the observer the crane boom must look little elongated. nut this length elongation occurs only in the horizontal direction. Thus, the horizontal component of the boom is

= 2.9 m

now from length contraction formula,

= 2.9 /(0.510)

= 5.68 m

this is the horizontal component of the boom, actual length is 5.68/ 0.5 = 11.36 m

here the angle remains the same since the moving frame is the car and not the boom.

c) Using Lorentz velocity transformation,

here u = velocity of the B with respect to earth i.e. velocity of the moving frame containing rocket B

u' = velocity of the rocket B with respect to the rocket A

v= velocity of rocket B with respect to A.

upon subs,

0.45 c + 0.77 c /(1 + (0.45c x 0.77c )/c²

= 1.22 c/ 1.3465

= 0.90 c

this is the velocity of rocket B wrt earth