An actor in an action movie is supposed to jump up and catch a ring moving overhead which will carry him to safety. The ring is 8 m above the actor and moving at 12 m/s to the right. The actor jumps straight up and catches the ring at the peak of his jump. How far to the left of the actor is the right when he must jump?

To find the distance to the left of the actor where the ring is when he must jump, we can use the concept of relative motion.

First, let's define a reference frame. We'll consider the actor as the origin of our coordinate system, and we'll assume the positive x-direction is to the right.

Given:
- The ring is 8 m above the actor.
- The ring is moving at a velocity of 12 m/s to the right.

Since the actor jumps straight up, his vertical velocity doesn't affect the horizontal motion of the ring. Therefore, we only need to consider the horizontal motion of the ring.

To find the distance the ring travels horizontally by the time the actor jumps, we can use the formula:

distance = velocity × time

In this case, we want to find the time it takes for the actor to reach the peak of his jump. Once we have the time, we can calculate the horizontal distance the ring travels during that time.

To find the time it takes for the actor to reach the peak of his jump, we need to know the actor's vertical motion. We'll assume that the actor jumps with a constant upward acceleration.

Using the equation of motion for vertical motion, we have:

Displacement = Initial Velocity × Time + (1/2) × Acceleration × Time^2

The actor's initial vertical velocity is 0 (as he starts from rest), and the displacement is 8 m (the height of the ring above the actor).

Therefore:

8 = 0 × Time + (1/2) × Acceleration × Time^2

Simplifying this equation, we can solve for the time it takes for the actor to reach the peak of his jump.

Now, once we have the time, we can calculate the distance the ring travels horizontally during that time using:

distance = velocity × time

Substituting the given values, we have:

distance = 12 m/s × time

Solve these equations to find the time and distance.