Physics
Physics
1st Edition
Walker
ISBN: 9780133256925
Textbook solutions

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Page 975: Standardized Test Prep

Exercise 1
Step 1
1 of 2
By definition of general theory of relativity.
Result
2 of 2
(C)
Exercise 2
Step 1
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They are related through Lorentz transformations.
Result
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(D)
Exercise 3
Step 1
1 of 2
We know that the time dilation formula is given by:

$$
begin{align*}
triangle t=frac{triangle t_0}{sqrt{1-frac{v^2}{c^2}}}
end{align*}
$$

where $triangle t$ is the time measured in the lab frame of reference, $triangle t_0$ is the proper time, $v$ is the speed of the moving frame of reference and $c$ is the speed of light in vacuum. We see from the equation above that the time measured in the laboratory frame of reference $triangle t$ is bigger than the proper time $triangle t_0$ (this is why the effect is called time dilation). We conclude that the right answer is: B). The moving clock must run faster, ie. it will measure a greater time interval than that of a clock that is stationary.

Result
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$$
B)
$$
Exercise 4
Step 1
1 of 2
Time dilation gives

$$
t=frac{1text{ s}}{sqrt{1-0.8^2}} =1.67text{ s}.
$$

Result
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(D)
Exercise 5
Step 1
1 of 2
Momentum goes to infinity as $v$ approaches $c$ so the correcti figure is $(B)$
Result
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(B)
Exercise 6
Step 1
1 of 2
Time intervals between the same events are different for observers in relative motion as said in (C)
Result
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(C)
Exercise 7
Step 1
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To calculate this we have to solve for $v$ in terms of $c$ the equation

$$
5text{ cm} = 10text{ cm}sqrt{1-frac{v^2}{c^2}}
$$

which gives

$$
v=csqrt{frac{3}{4}} = 0.87c.
$$

Result
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(B)
Exercise 8
Step 1
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a) Using relativistic expression we get

$$
E_1 = frac{mc^2}{sqrt{1-frac{v_1^2}{c^2}}} = 2.25times 10^{17} text{ J}.
$$

$$
E_2 = frac{mc^2}{sqrt{1-frac{v_2^2}{c^2}}} = 2.6times 10^{17}text{ J}.
$$

b) Using classical expression we get

$$
KE_1 = frac{1}{2}mv_1^2 =12500text{ J}
$$

$$
KE_2 = frac{1}{2}mv_2^2 = 2.81times 10^{16}text{ J}.
$$

c) Adding rest energy to classically calculated kinetic energy gives

$$
E_1’=2.25times10^{17}text{ J}
$$

$$
E_2′ = 2.53times10^{17}text{ J}.
$$

We see perfect matching for $v_1$ of the total energies and not so well matching for $v_2$. This is significant because it tells us that classical and relativistic expressions for speeds much smaller than $c$ give exactly the same results.

Result
2 of 2
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