# Time reversal in simple *solution* to equation of motion

Consider the solution to the equation of motion for a particle with a constant acceleration: $$x(t) = x_0 + v_0t + \frac{1}{2}at^2.$$

If I let $t \rightarrow -t$, then the equation becomes: $$x(-t) = x_0 - v_0t + \frac{1}{2}at^2,$$

which is different. Does this mean that this equation is not symmetric under time-reversal? What is the physical meaning of this? What if this represented a ball falling under gravity being recorded on tape: surely we should see the same thing if we run the film in reverse?

• this is not the equation of motion, but a solution to some equation. Jan 6, 2015 at 16:24
• That's not the equation of motion but a solution to the equation of motion. The equation of motion would be $x''(t)=a$. Jan 6, 2015 at 16:25
• Right, sorry, I'll change it Jan 6, 2015 at 16:25
• In normal time the ball speeds up. In reverse time it slows down. Of course the solution is not symmetric under time reversal. Jan 6, 2015 at 16:27
• But if $v_0=0$ then the equation is symmetric Jan 6, 2015 at 16:28

If you substitute $t\to-t$, the sign of the velocity also changes, thus the equation maintains the same functional form
you are correct that all of this time-reversal business starts with the mapping $t \mapsto -t$. But when you do this, you have to time-reverse EVERYTHING.
Since $v = \frac{\mathrm dx}{\mathrm dt}$, we need to reverse the time in the denominator, which gives us $v \mapsto \frac{\mathrm dx}{\mathrm d(-t)} = -v$
The acceleration is unchanged, since $$a \mapsto \frac{\mathrm d}{\mathrm d(-t)} \frac{\mathrm dx}{\mathrm d(-t)} = (-1)^{2}a = a$$