Question

Learning Goal: To practice Problem Solving Strategy 37.3 Lorentz Transformations. An enemy spaceship is moving toward your starfighter with a speed of 0.400 c , as measured in your reference frame. The enemy ship fires a missile toward you at a speed of 0.700 c relative to the enemy ship. If you measure the enemy ship to be 1.00×107 km away from you when the missile is fired, how much time t, measured in your frame, will it take for the missile to reach you?

Part D

Suppose that the missile has an internal clock that is turned on when the missile is fired. What time t∗ would the clock show just before the missile hits your startfighter?

Express your answer in seconds to three significant figures.

Answer 1

Given that :

speed of starfighter, u_s = 0.4 c

speed of enemy spaceship, v_s = 0.7 c

The starfighter is at distance, x = 1 x 10⁷ km = 1 x 10¹⁰ m

Time taken for the missile to reach which is given as :

using an equation,                   V_x = v_s + u_s / [1 + (v_s u_s/c²] { eq.1 }

inserting the values in above eq.

V_x = [(0.7 c) + (0.4 c)] / [1 + (0.7 c) (0.4 c) / c²]

V_x = (1.1 c) / (1.28)

V_x = 0.859 c

we know that,    x = V_x t                                                                           { eq.2 }

inserting the values in eq.2,

(1 x 10¹⁰ m) = (0.859 x 3 x 10⁸ m/s) t

t = (1 x 10¹⁰ m) / (2.57 x 10⁸ m/s)

t = 38.9 sec

Part-D : Suppose that the missile has an internal clock that is turned on when the missile is fired.

Time taken by the clock show just before the missile hits startfighter which will be given as :

using an equation, we have

t = t* / 1 - V_x² / c²

Or      t* = t 1 - V_x² / c²    { eq.3 }

inserting the values in eq.3,

t* = (38.9 sec) 1 - [(0.859 c)² / c²

t* = (38.9 sec) 1 - (0.737)

t* = (38.9 sec) 0.263

t* = (38.9 sec) (0.512)

t* = 19.9 sec