A 250 g object with a speed v of 7.6 m/s strikes a steel plate at an angle θ of 30° and rebounds with the same speed and angle . It is in contact with the wall for 11 ms. What is the average force on the wall from the ball?

To find the average force exerted on the wall by the object, we need to analyze the momentum change during the collision. The key here is to understand how the object's momentum changes when it strikes the wall and then rebounds.

Understanding Momentum

Momentum is defined as the product of an object's mass and its velocity. The formula for momentum (p) is:

p = m × v

Where:

• m is the mass of the object (in kilograms)
• v is the velocity of the object (in meters per second)

Calculating Initial and Final Momentum

In this scenario, the object has a mass of 250 g, which we convert to kilograms:

m = 250 g = 0.25 kg

The speed of the object is given as 7.6 m/s, and it strikes the wall at an angle of 30°. We can break down the velocity into its horizontal and vertical components using trigonometric functions:

• Vx = v × cos(θ)
• Vy = v × sin(θ)

Calculating these components:

Vx = 7.6 m/s × cos(30°) ≈ 7.6 m/s × 0.866 = 6.58 m/s

Vy = 7.6 m/s × sin(30°) = 7.6 m/s × 0.5 = 3.8 m/s

Initial Momentum Before Impact

The initial momentum vector before the impact can be expressed as:

p_initial = m × Vx (horizontal) + m × Vy (vertical)

However, since the wall only affects the horizontal component, we focus on:

p_initial_x = m × Vx = 0.25 kg × 6.58 m/s ≈ 1.645 kg·m/s

Final Momentum After Rebound

After rebounding, the horizontal component of the velocity will be in the opposite direction, but the speed remains the same:

p_final_x = m × (-Vx) = 0.25 kg × (-6.58 m/s) ≈ -1.645 kg·m/s

Change in Momentum

The change in momentum (Δp) during the collision can be calculated as:

Δp = p_final - p_initial

Substituting the values:

Δp = -1.645 kg·m/s - 1.645 kg·m/s = -3.29 kg·m/s

Calculating Average Force

To find the average force (F_avg) exerted on the wall, we can use the impulse-momentum theorem, which states that the impulse (change in momentum) is equal to the average force multiplied by the time of contact (Δt):

F_avg = Δp / Δt

Given that the time of contact is 11 ms (which we convert to seconds):

Δt = 11 ms = 0.011 s

Now we can calculate the average force:

F_avg = -3.29 kg·m/s / 0.011 s ≈ -299.09 N

Final Result

The negative sign indicates that the force is in the opposite direction of the initial momentum. Therefore, the average force exerted on the wall by the ball is approximately:

299.09 N

This force acts perpendicular to the wall, reflecting the change in momentum as the ball rebounds off the surface. Understanding these principles helps us analyze collisions in various physical contexts, from sports to engineering applications.