Problem 353 - JEE Physics

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) \(12 \mathrm{~N}\)

Step-by-Step Solution

Answer

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

Step 1: 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: \(p_{y1} = mv\sin(\theta)\)

Step 2: 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: \(p_{y2} = mv\sin(\theta)\)

Step 3: 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: \(\Delta p_y = p_{y2} - p_{y1} = mv\sin(\theta) - mv\sin(\theta) = 0\)

Step 4: 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: \(F_{avg} = \frac{\Delta p_x}{\Delta t}\)

Step 5: 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).

Key Concepts

Collision mechanics

When objects collide, various factors come into play to determine the outcome of the collision. In our example, a ball hits a wall, showcasing an example of an elastic collision where the speed before and after the collision remains the same, despite direction changes. Here's what happens in a collision like this:

• Elastic collision: Both kinetic energy and momentum are conserved. In this case, the ball bounces back with the same speed it approached the wall.
• Change of direction: Even if the speed stays the same, the change in direction needs examination since it affects the momentum components.

Collision mechanics help us understand how forces and movement interact when two bodies meet, showing the usefulness of physics principles in predicting and analyzing physical events.

Change in momentum

Momentum and its change is a central concept to understanding collisions. Defined as the product of an object's mass and velocity, momentum measures how difficult it is to stop a moving object. Mathematically, momentum is expressed as \( p = mv \).
For the ball hitting the wall, its velocity changes direction after the collision. This change only affects the horizontal component of momentum:

• Initial Momentum: Before hitting the wall, the ball has two momentum components: horizontal and vertical.
• Final Momentum: After the collision, the ball's horizontal component of momentum reverses direction, while the vertical component stays the same.
• The change in momentum is crucial as it defines the force exerted during the collision.

Recognizing how momentum changes during interactions helps us calculate forces involved in real-world collision occurrences.

Average force calculation

The average force can be computed by examining the change in momentum over the time for which the force acts. Using Newton's Second Law, we know that force is related to the change in momentum and can be described by the formula:\[ F_{avg} = \frac{\Delta p}{\Delta t} \]Here's how it's applied:

• Identify change in momentum: This is calculated using both initial and final momentums and determines the impact's effect.
• Select the time interval: In this case, it's the time the ball stays in contact with the wall, a key factor in measuring average force.

Understanding how to calculate average force gives us insight into how forces act over time, informing both engineering and everyday problem-solving.

Impact force

Impact force is specifically the force generated by two objects colliding in a very short time frame. In high-speed transactions, this force can be quite substantial. It derives from a consideration of how quickly momentum must change during the collision.
In our ball and wall scenario, calculating impact force involves:

• Change in momentum: Knowing how the momentum shifts due to collision informs the force magnitude.
• Duration of impact: The short contact time between the ball and the wall influences the size of the force.

Understanding impact force helps in designing safer vehicles, sports gear, and other equipment where collisions are likely, ensuring that safety and function go hand in hand.