Answer
In both cases, a speed of about 0.14c results in a 1 percent relativistic effect.
Work Step by Step
a. Solve for the speed when the length is 99 percent of the rest length, using the length contraction relationship, equation 26–3a.
$$\mathcal{l} = \mathcal{l}_{o} \sqrt{1-\frac{ v^{2} } { c^{2} }} $$
$$v=c\sqrt{1-(\frac{\mathcal{l}}{\mathcal{l}_o})^2}$$
$$v=c\sqrt{1-(\frac{0.990}{1.00})^2}$$
$$v=0.141c $$
b. Solve for the speed when the dilated time is 101 percent of the proper time, using the time dilation relationship, equation 26–1a.
$$\Delta t_o = \Delta t \sqrt{1-\frac{ v^{2} } { c^{2} }} $$
$$v=c\sqrt{1-(\frac{\Delta t_o}{\Delta t})^2}$$
$$v=c\sqrt{1-(\frac{1}{1.01})^2}$$
$$v=0.140c $$
