Chapter 6: Q3E (page 1247)

How fast must a rocket travel relative to the earth so that time in the rocket “slows down” to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

Short Answer

Expert verified

u = 0.867c

Step by step solution

Concept of the time dilation equation

The time dilation equation is given as $∆ t = γ ∆ t_{0} / \sqrt{1 - u^{2} / c^{2}}$ where $∆ t_{0}$ is the proper time in the rest frame, $∆ t$ is the time interval measured in the second frame of reference u is the speed of the second frame with respect to the rest frame, c is the speed of light

Calculate the speed of the jet plane.

We know that, $Δ t = \frac{1}{2} Δ t_{0}$ In order to calculate the u of the rocket, we will use the time dilation equation: $Δ t = γ Δ t_{0}$

$y = \sqrt{1 - u^{2} / c^{2}} ∆ t = ∆ t_{0} / \sqrt{1 - u^{2} / c^{2}} \sqrt{1 - u^{2} / c^{2}} = ∆ t_{0} / ∆ t$

By squaring both sides:

$1 - u^{2} / c^{2} = {(\frac{1}{2})}^{2} u^{2} / c^{2} = 1 - {(\frac{1}{2})}^{2}$

By taking square root for both sides:

$\sqrt{u^{2} / c^{2}} = \sqrt{1 - {(\frac{1}{2})}^{2}} \frac{u}{c} = 0.867 u = 0.867 c$

The fastest jet plane is around $1.09 \times 10^{3} m / s$

Thus, there is no jet plane nowadays approaches such speeds.

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How fast must a rocket travel relative to the earth so that time in the rocket “slows down” to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

Short Answer

Step by step solution

Concept of the time dilation equation

Calculate the speed of the jet plane.

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