AP Physics 1 Exam Question:

A ball is thrown vertically upward with an initial velocity of 30 m/s from a height of 10 meters. The acceleration due to gravity is -9.8 m/s².

(a) What is the maximum height reached by the ball?

(b) How long does it take for the ball to reach its maximum height?

(c) How long does it take for the ball to fall back to the ground from its maximum height?

Solution:

(a) To find the maximum height reached by the ball, we can use the equation of motion:

                vf^2 = vi^2 + 2ad

where vf is the final velocity (0 at the highest point), vi is the initial velocity (30 m/s), a is the acceleration (-9.8 m/s²), and d is the displacement (maximum height).

Plugging in the known values, we get:

                0^2 = 30^2 + 2*(-9.8)*d

Simplifying the equation, we have:

                0 = 900 - 19.6d

Solving for d, we get:

                19.6d = 900
                d = 900 / 19.6
                d ≈ 45.92 m

Thus, the maximum height reached by the ball is approximately 45.92 meters.

(b) To find the time it takes for the ball to reach its maximum height, we can use the equation of motion:

                vf = vi + at

At the maximum height, the final velocity is 0 m/s. The initial velocity is 30 m/s, and the acceleration is -9.8 m/s².

Plugging in the known values, we get:

                0 = 30 + (-9.8)t

Simplifying the equation, we have:

                -30 = -9.8t

Dividing both sides by -9.8, we find:

                t = -30 / (-9.8)
                t ≈ 3.06 s

Thus, it takes approximately 3.06 seconds for the ball to reach its maximum height.

(c) To find the time it takes for the ball to fall back to the ground from its maximum height, we can use the equation of motion:

                d = vit + (1/2)at^2

where d is the displacement (the maximum height of 45.92 meters), vi is the initial velocity (0 m/s at the highest point), a is the acceleration (-9.8 m/s²), and t is the time.

Plugging in the known values, we get:

                45.92 = 0*t + (1/2)*(-9.8)*t^2

Simplifying the equation, we have:

                45.92 = -4.9t^2

Rearranging the equation, we find:

                t^2 = 45.92 / -4.9
                t^2 ≈ -9.38

Since time cannot be negative, we reject the negative solution. Thus, there is no real solution for t in this equation.

Therefore, the ball does not fall back to the ground from its maximum height, as it has reached its peak and is now on its downward trajectory.

Answer Summary: (a) The maximum height reached by the ball is approximately 45.92 meters. (b) It takes approximately 3.06 seconds for the ball to reach its maximum height. (c) The ball does not fall back to the ground from its maximum height.