Question

At what speed, in $m/s,$would a moving clock lose role="math" localid="1649531760117" $1.0\text{ns}$in $1.0$day according to experimenters on the ground?

Hint: Use the binomial approximation.

Short answer

The speed of a moving clock is $46m/s.$

Step-by-step solution

Given Information

We have to find the speed in$m/s$ for a moving clock lose $1.0\text{ns}$in $1.0$day according to experimenters on the ground using binomial approximation.

Simplify

The ground's frame is $S$.

The moving clock frame is $S\text{'}.$

$$\begin{gathered} ∆t=1.0d+1.0ns \\ ∆t\text{'}=1.0d=86400s \end{gathered}$$

We have: $∆t=y∆t\text{'}.$

When calculating powers of sums of 1 and a small number x, the binomial approximation comes in handy.

Now, we use binomial approximation for $y:$

localid="1649844482103" $$\begin{gathered} \frac{∆t}{∆t\text{'}}=1+\frac{v^2}{2c^2} \\ \frac{∆t}{∆t\text{'}}-1=\frac{v^2}{2c^2} \\ 2c^2\left(\frac{∆t}{∆t\text{'}}-1\right)=v^2 \\ v=c\sqrt{2\left(\frac{∆t}{∆t\text{'}}-1\right)} \\ v=c\sqrt{2\left(\frac{1.0ns}{1.0d}\right)} \\ v=(3\times {10}^{8}m/s)\sqrt{\left(\frac{2(1.0\times {10}^{-9}s)}{86400s}\right)} \\ v=46m/s \end{gathered}$$