Problem — Sci-fi physics, time reversed

Problem

Sci-fi physics: reversible time. A projectile in the air moves to the right at $45^\circ$. Discuss in five lines how the trajectory would change if time were reversed. What stays the same?

Solution

Reversing time means substituting $t \to -t$ in the equations of motion. The position $\vec r(t)$ depends on $t$ only through $t^2$ (the gravity term) and $t$ (the initial-velocity term), so the projectile retraces the same parabola in space, but in the opposite direction: it starts from the landing point and returns to the launch point.

The velocities change sign ($\vec v \to -\vec v$): where the projectile was rising before, it now descends, and the horizontal motion is to the left instead of to the right. The projectile launched at $45^\circ$ to the right becomes a projectile arriving at $45^\circ$ from the right.

What stays the same is the acceleration: $\vec g$ always points downward, because it depends on $t^2$ and the square does not change under the reversal. This is precisely what makes the trajectory an identical parabola: gravity keeps curving the motion downward in both directions of time.

Links

Topics: Cinematica Concepts: Moto parabolico Skills: Analisi di casi limite e fantafisica