Question

Lauren has a bow that can shoot arrows at speeds of 0.3c. She is travelling across the Great Sandy Desert to her next archery competition on an ultrafast train that moves at a speed of 0.5c. She has set up to practise on the train with a target that is 150 m away from her in the same direction that the train is travelling. Greg is standing on a platform as the train travels by.

(a) What is the time between the events “Lauren shoots arrow” and “arrow hits target” for Lauren?

(b) How far is Lauren from the target as measured by Greg?

(c) When Lauren shoots an arrow, what is its speed as measured by Greg?

(d) Do either Greg or Lauren measure the proper time between the events described in (a)? Explain your answer.

(e) How long does it take for the arrow to reach the target as measured by Greg?

Answer 1

speed of arrow with respect to lauren = = 0.3c

speed of train with respect to ground == 0.5c

c = speed of light = 3 x 10^8 m/s

When lauren is in the train, then velocity of lauren and arrow is also 0.5c

distance between lauren and target = 150 m.

a) relative velocity of arrow with respect to train(or lauren) is = 0.3c

therefore, time taken by arrow to reach from lauren to target = t = 150/0.3c = 0.000002 s. [answer]

b) As both the target and lauren is stationary with respect to train, which means both are moving with same velocity 0.5c .

then for greg, the separation will remain same i.e; 150 m [answer]

c) = 0.3c =

= 0.5c

velocity of arrow with respect to ground or greg. =

from definition of relativ emotion :

as both and are in same direction, [answer]

d) Time does not depend on anything, so for both observers the time will remain same.

e) for greg, both target and arrow is moving.

velocity of target = = 0.5c

velocity of arrow is = 0.8c

if the target travels x distance in time t, then arrow travells 150+x distance in time t.

therefore,

for target : x = * t

for arrow : 150 + x = * t

=> 150 + ( * t) =   * t

=> t = 150 / ( - ) = 150 / (0.8c - 0.5c) = 150 / 0.3c =  0.000002 s [answer]