A 2.25 kg ball experiences a net force of 965 N up a ramp. Once the ball reaches the top of the ramp, the force no longer acts. The force acts over a distance of 1.50 m on the ramp. Find the horizontal distance x that the ball travels before it hits the deck. The top of the ramp is 4.50 meters above the deck below.

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To find the horizontal distance the ball travels before hitting the deck, we can break down the problem into two stages: the motion up the ramp and the motion back down to the deck.

First, let's analyze the motion up the ramp:

1. Identify the known values:
- Mass of the ball (m) = 2.25 kg
- Net force up the ramp (F) = 965 N
- Distance over which the force is applied (d) = 1.50 m
- Height of the top of the ramp (h) = 4.50 m

2. Calculate the work done on the ball:
Work (W) = Force × Distance
W = F × d

In this case, since the force is acting opposite to the displacement (up the ramp), the work done is negative:
W = -965 N × 1.50 m

3. Calculate the gravitational potential energy change of the ball:
Gravitational potential energy change (∆PE) = m × g × h
∆PE = 2.25 kg × 9.8 m/s² × 4.50 m

Since the ball starts at rest, the initial kinetic energy (KE) is zero.

4. Use the work-energy theorem to relate work and energy:
Work (W) + ∆KE = ∆PE

Since the initial kinetic energy is zero, the equation simplifies to:
W = -∆PE

Now we can equate the two expressions for work:
-965 N × 1.50 m = -∆PE
Solve for the potential energy change (∆PE).

5. Once you find the potential energy change (∆PE), equate it to the gravitational potential energy change:
∆PE = m × g × h
Solve for the gravitational acceleration (g).

Next, let's analyze the motion down to the deck:

6. Use the horizontal distance formula:
h = x + √(d² - x²)

Given that h = 4.50 m and d = 1.50 m, solve for x using the equation above.

Now you have the horizontal distance x that the ball travels before hitting the deck.