A ball starts from rest at the initial position x; = 0. The ball has a constant acceleration of 2.4 m/s2. (a) Write
the position-time equation for the ball. (b) What is the
position of the ball at t = 1.0 s? (c) What is its position
at t = 2.0 s?

We want to get the motion equations of a ball, and we want to use them to find the position of the ball at different times.

The solutions are:

We know that the ball starts from rest (so the initial velocity is zero) from the position x = 0.

The ball has a constant acceleration of 2.4 m/s^2

Then the acceleration equation is:

A(t) = 2.4 m/s^2

To get the velocity equation we integrate, remember, the initial velocity is zero, so we don't have a constant of integration:

V(t) = (2.4 m/s^2)*t

To get the position equation we integrate again, and because the initial position is x = 0, we don't have a constant of integration:

P(t) = 0.5*(2.4 m/s^2)*t^2

a) The position equation is:

P(t) = (1.2 m/s^2)*t^2

b) The position of the ball at t = 1s is given by:

P(1s) =  (1.2 m/s^2)*(1s)^2 = 1.2m

The ball position at t = 1s is x = 1.2m

c) Similar to the above case:

P(2s) = (1.2 m/s^2)*(2s)^2 = 4.8 m

x = 4.8m

If you want to learn more, you can read:

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the independent variable, the one that is intentionally manipulated, is whether or not the car hits the barrier.

the dependent variable, the one that you measure the response in, is the distance that the washers travel.