### Video Transcript

If relativistic effects are to modify time and length values by less than one percent then the Lorentz factor, πΎ, must be less than one point zero one. At what relative velocity is πΎ equal to one point zero one?

Letβs call the relative velocity weβre solving for π£. We want to find that velocity π£ such that πΎ is one point zero one. Weβll start by recalling the mathematical expression for the Lorentz factor, πΎ.

The Lorentz factor, πΎ, is defined as one divided by the square root of one minus π£ squared over π squared. π£ represents our relative speed, and π is the speed of light.

Applying this relationship to our scenario, we have been given πΎ, thatβs one point zero one, and we want to solve for π£, the relative velocity, at which πΎ has their value.

To do that letβs rearrange this Lorentz factor equation. We start by squaring both sides. Looking at the right-hand side, since one squared is one, the right side of our equation becomes one over one minus π£ squared over π squared.

Next we can multiply both sides of the equation by the quantity one minus π£ squared divided by π squared, which cancels that entire term from the right-hand side of our equation.

So we now have an equation that reads the quantity one minus π£ squared over π squared times πΎ squared is equal to one. Multiplying through by πΎ squared on the left-hand side then adding π£ squared divided by π squared times πΎ squared to both sides cancels that term out on the left side of the equation.

And then subtracting one from each side cancels that one from the right side. Weβre left with an equation that reads πΎ squared minus one is equal to π£ squared divided by π squared times πΎ squared.

Our next algebraic step is to multiply both sides of the equation by π squared divided by πΎ squared, which cancels the πΎ squared and π squared terms from the right-hand side of equation.

Finally we take the square root of both sides. This cancels out the square and square root terms on the right side, leaving us with a simplified equation for the relative velocity π£.

π£ is equal to the speed of light π divided by the Lorentz factor πΎ times the square root of πΎ squared minus one. Rather than give our final answer for π£ in units of meters per second, weβll give it in terms of the speed of light π.

We can now plug in for the value of πΎ, one point zero one. The relative velocity when πΎ is one point zero one is equal to zero point one four zero times π, the speed of light.

This is the relative velocity that matches a Lorentz factor of one point zero one.