Problem 338 - JEE Physics

Question

A player caught a cricket ball of mass \(150 \mathrm{~g}\) moving at the rate of \(20 \mathrm{~ms}^{-1}\). If the catching process be completed in \(0.1 \mathrm{~s}\) the force of the blow exerted by the ball on the hands of player is (A) \(0.3 \mathrm{~N}\) (B) \(30 \mathrm{~N}\) (C) \(300 \mathrm{~N}\) (D) \(3000 \mathrm{~N}\)

Step-by-Step Solution

Answer

The magnitude of the force exerted by the ball on the player's hands is \(30 \mathrm{~N}\).

Step 1: Convert mass to kilograms.

First, we need to convert the mass of the ball from grams to kilograms, since all other units are in the SI system. To do this, just divide the given mass by 1000: \[ mass = \frac{150~g}{1000} = 0.15~kg \]

Step 2: Calculate the initial momentum of the ball.

We'll call the initial velocity of the ball \(v_i =20~m/s\). Then, we can calculate the initial momentum of the ball as: \[ p_i = (mass)(v_i) = (0.15~kg)(20~m/s) = 3~kg \cdot m/s \]

Step 3: Determine the final momentum of the ball.

Since the ball is caught and comes to a stop, its final velocity is zero (\(v_f = 0~m/s\)). Therefore, the final momentum of the ball is also zero: \[ p_f = (mass)(v_f) = (0.15~kg)(0~m/s) = 0~kg \cdot m/s \]

Step 4: Calculate the change in momentum, or impulse.

The change in momentum, also known as impulse, can be calculated as the difference between the final and initial momenta: \[ \Delta p = p_f - p_i = 0~kg \cdot m/s - 3~kg \cdot m/s = -3~kg \cdot m/s \]

Step 5: Use Newton's second law of motion to calculate the force.

Newton's second law of motion states that the force (\(F\)) acting on an object is equal to the change in momentum (\( \Delta p\)) divided by the time interval (\( \Delta t\)): \[ F = \frac{\Delta p}{\Delta t} \] Now we'll plug in the values we've calculated to find the force exerted by the ball on the hands of the player: \[ F = \frac{-3~kg \cdot m/s}{0.1~s} = -30~N \] As the force is negative, it means that the force exerted by the ball on the hands of the player is in the opposite direction of the ball's initial velocity. So the magnitude of the force: \[ |F| = 30 ~N \] Taking the magnitude of the force, the correct answer is (B) \(30 \mathrm{~N}\).

Key Concepts

Impulse and Momentum

When a cricket ball is caught, it undergoes a significant change in momentum. Momentum is defined as the product of an object's mass and velocity, expressed mathematically as:\[ p = mv \]In this exercise, the ball has an initial velocity and mass that contribute to its momentum. Impulse, on the other hand, represents the change in an object's momentum. It's important to note that impulse can result from a force acting over a certain time interval.

• Initial momentum can be calculated if the mass and velocity of the object are known.
• Final momentum accounts for any changes in speed or when an object comes to a stop, as seen when the cricket ball is caught.
• The impulse is the net change between initial and final momentum.

In this scenario, since the ball is caught and brought to a stop, its final momentum is zero. Calculating the impulse involves subtracting this zero final momentum from the initial momentum, which reflects the magnitude of change that occurred.

Force Calculation

Force is a fundamental concept in Newtonian physics, explaining how motion changes when introduced or resisted. According to Newton's second law, force is quantified as the change in momentum over time:\[ F = \frac{\Delta p}{\Delta t} \]In this problem, once we've calculated the change in momentum (or impulse), the next step is calculating the force exerted by the cricket ball as it is caught by the player. This force is computed over a given time interval during which the catching process occurs.

• Identify the time interval to understand the duration of force application.
• Use impulse values to compute the force, keeping in mind that time must be converted to seconds if it isn't already.
• A negative force value typically indicates direction relative to motion. However, the magnitude of force refers to the absolute value.

The calculated force helps determine how much impact the player experiences, showing the effectiveness of Newton's laws in describing everyday activities.

SI Units Conversion

In physics, consistency in units is crucial for accurate calculations. The International System of Units (SI) is globally accepted, facilitating a coherent method to quantify physical quantities. This exercise highlights the importance of converting units for mass, particularly moving from grams to kilograms.

• Mass in grams must be converted to kilograms to align with standard SI units.
• This conversion typically involves dividing the mass value by 1000, which is simple yet vital for accurate calculations.
• Velocity and time should ideally be in meters per second and seconds, respectively, to ensure uniformity.

Adhering to SI units across all calculations yields reliable results and seamless integration into the broader physics context, which is foundational when applying Newton's laws of motion.