What Is the Time Interval Between Photon Pulses Observed from Earth?

[Delzac]

Homework Statement

An enemy spaceship moves past the Earth with a speed of 0.8 c. The captain orders the spaceship weapons department to blast the Earth with pulsed laser photons every 10 seconds. For the observers on Earth who see the flashes, what is the time interval they measure between photon pulses?

Homework Equations

$$\Delta T = \gamma \Delta T_0$$

The Attempt at a Solution

Since spaceship is moving,

$$\Delta T_0 = 10s$$
$$v = 0.8c$$
subbing in the values into the formula we get :
$$\Delta T = 16.7s$$

Is this correct?

Another Qns :

for the formula $$\Delta T = \gamma \Delta T_0$$
As a guide line, is $$\Delta T > \Delta T_0$$ for all cases?
Also, does the $$\Delta T_0$$ mean the duration in which the event happen in the frame of the event. Which in this case is the 10s.

Any help will be appreciated

 

[Hootenanny]

$$\Delta T_0 = 10s$$
$$v = 0.8c$$
subbing in the values into the formula we get :
$$\Delta T = 16.7s$$

Is this correct?

Yes, this is correct.

Another Qns :

for the formula $$\Delta T = \gamma \Delta T_0$$
Also, does the $$\Delta T_0$$ mean the duration in which the event happen in the frame of the event. Which in this case is the 10s.

T₀ is what is known as proper time. Proper time is the time interval measured in the rest frame of an event(s). An alternative definition of proper time is "when two events occur at the same location in inertial reference frames, the time interval between them, measured in that frame, is called the proper time interval"

As a guide line, is $$\Delta T > \Delta T_0$$ for all cases?

Yes, this is true for ALL cases, hence the name time dilation. Let us examine the formula;

$$\Delta t = \gamma\Delta t_{0}=\frac{\Delta t_{0}}{\sqrt{1-\beta^2}}\hspace{1cm}\beta = \frac{v}{c}<1$$

Since the ratio $\beta$ must be less than one (v<c), it follows that the denominator of the above equation must also be less than one and hence $\Delta t$ must always be greater than $\Delta t_0$ .

 

[Delzac]

k thanks for the help.

 

[MeJennifer]

It is true that there is a time dilation because of the speed differential between the spaceship and the earth. So for that situation you can use the Lorentz factor.

But the moment of firing is also important. When the spaceship approaches the Earth the time interval between potential blasts dilates less and when the spaceship moves away from the Earth the interval dilates more. Since the problem statement says that the ship moves past the Earth and then starts to blast, if moves away, so the interval dilates even more.

 

[Hootenanny]

Good point Jennifer, I didn't think of that.