Two spaceships U (unprimed) and P (primed) have identical proper lengths of 120m. The two are passing near each other with relative speed of u=0.8c (see figure 1). U has a single gun at its rear and P has two guns, one at 40m from the front and the other at 40m from the rear. According to the code of the galaxy, a spaceship can signal its peaceful intention by firing a missile which is timed to miss the other ship. Wishing to do this, the pilot of U fires its gun when the front of U coincides with the rear of P. The pilot of U knows that P will be relativistically contracted and the missile from U will not hit P(see figure 2). We assume that the two pass closely enough so that the travel time of the missile is small enough to be completely ignored. However, from the point of P, it is U that is relativistically contracted and so the missile is sure to strike P (see figure 3). Should P fire its guns for protection?
For the following calculations, use c=30 m/s for the speed of light. This will make the calculations more manageable by avoiding cumbersome arithmetic. Substitute the given numerical values for u and c into the Lorentz transformations and write down the equations in final form, ready to be used for length contraction and time dilation computations.
Each ship is located in its own frame of reference as follows:
Front of U at 300m and rear of U at 420m
Front of P at 220m and rear of P at 100m.
To help solve the problem, make the three "photo-montages" described below. Draw each using the scale mm = 1m of proper length.
1. Show the situation as seen by U when the front coincides with rear of P .
2. Show the same situation as seen by P.
3. Show the situation as seen by P when the gun on U fires.
Make your drawings ( the direction in which each ship moves, etc.) resemble the sample drawing as much as possible. Each photo-montage must contain the space and time coordinates such as x,t or x',t' for
a) the front and rear of U,
b) the front and rear of P, (in your drawing of #3 photo- montage, to show only the front half of P).
c) locate the gun on U and indicate whether it is yet to fire, is just firing, or has already been fired.
4. Should P fire its guns for protection ? Explain, drawing inferences from the results of your calculation and the photo-montages.
P is feverishly at work with the calculation of Lorentz transformations, trying to determine whether the missile from U will hit or miss. Unfortunately, the calculator battery is close to depletion and it makes an error in the computation which makes P to believe that U is
unfriendly. When the centers of both the ships are opposite each other, P
fires both guns simultaneously.
5. Make a fourth photo-montage showing the situation as seen by P when the centers of both ships coincide and the guns are just fired. Include the space and time coordinates of the following :
a. the centers of U and P
b. the front and rear of U
c. the front and rear of P
d. location of all guns
e. guns on P firing
Refer to the fourth photo-montage for the following.
6. Where do the missiles hit U ?
7. How far apart are the hits on U ?
8. How far apart are the guns on P ?
9. Compare 7. and 8. and explain from the view point of both U and P.
Luckily, U has recently acquired a missile neutralizer. These being rather expensive, U had to settle for a single missile model, which must be placed at the site of the missile impact. U tries to reason that, since P fired the missiles simultaneously, U will not see them fire simultaneously but in succession. So the hope of moving the single missile neutralizer to wipe off the two hits from P to protect itself, lingers on.
10. What time does U see each missile fire ?
11. Where and when should U first place the neutralizer ? Where and when should U next place it ?
12. How fast (assume constant speed) must U then move the neutralizer to reach the impact point of the other missile in time to neutralize it ?
13. Can U succeed ? (The number of this question is a clue!) Explain from the view point of both U and P.
It would be instructive to check some lengths and time intervals, obtained from pairs of events in your drawings, against the values calculated using the length contraction and time dilation formulas. Comparisons like these help you learn how to properly use these formulas.
For your practice, verify the space and time coordinates of a typical photo-montage shown below. It is the view as seen by U when the rear of U is opposite the front of P. The event E1 that begins construction of the montage is : x1= 420m (rear of U) and x'= 220m (front of P) are at the same position in space. This information then determines the values of t1 and t1'. E1 is indicated in the drawing by a star. Then, since this is a view as seen by U, all times in the U system must be the same as that of E1.
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Updated 01/24/00