Question

A \(0.5 \mathrm{~kg}\) ball moving with a speed of \(12 \mathrm{~ms}^{-1}\) strikes a hard wall at an angle of \(30^{\circ}\) with the wall. It is reflected with the same speed and at the same angle. If the ball is in contact with the wall for \(0.25 \mathrm{~S}\) the average force acting on the wall is (A) \(96 \mathrm{~N}\) (B) \(48 \mathrm{~N}\) (C) \(24 \mathrm{~N}\) (D) \(12 \mathrm{~N}\)

Short answer

The average force acting on the wall is \(48 \mathrm{~N}\), which corresponds to option (B).

Step-by-step solution

Calculate the initial horizontal and vertical components of the momentum

First, we need to calculate the initial momentum components of the ball. The vertical component remains unchanged, while the horizontal component changes its direction after the collision. To calculate the components of the momentum, we use the mass of the ball and its initial speed, along with the given angle. Initial horizontal momentum: \(p_{x1} = mv\cos(\theta)\) Initial vertical momentum: \(p_{y1} = mv\sin(\theta)\)

Calculate the final horizontal and vertical components of the momentum

After the collision, the ball maintains its speed, and therefore, the vertical component of the momentum remains unchanged, while the horizontal component changes direction. We can calculate the final components of the momentum using the same formulas as before. Final horizontal momentum: \(p_{x2} = -mv\cos(\theta)\) (negative sign because the direction is opposite to the initial direction) Final vertical momentum: \(p_{y2} = mv\sin(\theta)\)

Calculate the change in the momentum of the ball

To calculate the change in the momentum, we will subtract the initial momentum components from the final momentum components. Change in horizontal momentum: \(\Delta p_x = p_{x2} - p_{x1} = -mv\cos(\theta) - mv\cos(\theta) = -2mv\cos(\theta)\) Change in vertical momentum: \(\Delta p_y = p_{y2} - p_{y1} = mv\sin(\theta) - mv\sin(\theta) = 0\)

Calculate the average force exerted on the wall

Now, we will calculate the average force exerted on the wall by the ball. For this, we will use the change in horizontal momentum and the time for which the ball is in contact with the wall. Average force: \(F_{avg} = \frac{\Delta p_x}{\Delta t}\)

Plug in the given values and solve for the average force

Now that we have the formula for the average force, let's plug in the given values and solve for it. \(F_{avg} = \frac{-2(0.5 \mathrm{~kg})(12 \mathrm{~ms}^{-1})\cos(30^{\circ})}{0.25 \mathrm{~S}}\) Calculating this expression, we get: \(F_{avg} = 48 \mathrm{~N}\) The average force acting on the wall is 48 N, which corresponds to option (B).