Problem 47 - Master Resource Book in JEE Main Physics

Question

A boy released a ball from the top of a building. It will clear a window \(2 \mathrm{~m}\) high at a distance \(10 \mathrm{~m}\) below the top in nearly (a) \(1 \mathrm{~s}\) (b) \(1.3 \mathrm{~s}\) (c) \(0.6 \mathrm{~s}\) (d) \(0.13 \mathrm{~s}\)

Step-by-Step Solution

Answer

The ball clears the window in 1.3 seconds (option b).

Step 1: Understand the Problem

A ball is released from rest, and the task is to calculate the time it takes for the ball to clear a window located 10 meters below the top of the building. The height of the window is 2 meters.

Step 2: Use the Kinematic Equation

We need to use the kinematic equation for an object initially at rest under the influence of gravity: \( h = \frac{1}{2} g t^2 \). Here, \( h \) is the vertical distance (10 m), and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²).

Step 3: Solve for Time

Substitute the known values into the equation: \( 10 = \frac{1}{2} \times 9.81 \times t^2 \). Solving for \( t \): \[ t^2 = \frac{10 \times 2}{9.81} \]\[ t = \sqrt{\frac{20}{9.81}} \]

Step 4: Calculate the Time

Perform the calculations: \[ t = \sqrt{\frac{20}{9.81}} \approx 1.43 \text{ s} \]

Step 5: Choose the Closest Option

Comparing the calculated time 1.43 s with the given options, the nearest choice is (b) 1.3 s.

Key Concepts

Kinematic Equations

Kinematic equations are fundamental tools in physics that describe the motion of objects. They are particularly useful when analyzing projectile motion, like in the case of a ball in free fall. Kinematic equations relate several key parameters of motion: initial velocity, final velocity, acceleration, time, and displacement. For an object starting from rest under constant acceleration, like gravity, the main equation we often use is: \[ h = \frac{1}{2} g t^2 \]Here,

• \( h \) is the displacement or height the object travels,
• \( g \) is the acceleration due to gravity,
• and \( t \) is the time elapsed.

This formula allows us to calculate the time it takes for an object to fall a certain distance. In our problem, the ball falls 10 meters, and we use this equation to find the time it takes to reach that point. Breaking down the kinematic equation simplifies our calculations.By rearranging the equation to solve for \( t \), we get \( t = \sqrt{\frac{2h}{g}} \). Substituting the given heights and gravitational acceleration, we determined the time taken by the ball to clear the window.

Acceleration due to Gravity

Gravity is a universal force that attracts two bodies towards one another. On Earth, it provides a constant acceleration to objects in free fall. This acceleration due to gravity, denoted as \( g \), is approximately \( 9.81 \, \text{m/s}^2 \) near the Earth's surface.This constant value means that any object in free fall, regardless of its mass, will experience the same acceleration.

• This principle simplifies many projectile motion problems.
• We assume that air resistance is negligible, focusing purely on gravitational effects.

In our exercise, the ball is released from rest. The natural force of gravity accelerates the ball as it falls. This acceleration is crucial to predicting the ball's behavior over time. Since the ball starts from a state of rest, its initial velocity is zero, and gravity does all the work as it falls.Understanding this concept helps us use the kinematic equation correctly, ensuring we can calculate how long it takes for the ball to reach different points during its descent.

Free Fall

Free fall describes the motion of an object under the influence of gravitational force only. An object in free fall experiences constant acceleration due to gravity, with no other forces acting upon it.When a ball is released, like in our exercise, it enters a state of free fall. Key aspects of free fall include:

• Constant acceleration: The only acceleration is due to gravity, \( g = 9.81 \, \text{m/s}^2 \).
• Initial velocity is zero when dropped: The ball starts from rest.
• Increasing velocity: As time progresses, the velocity of the falling ball increases linearly because of the constant acceleration.

This means that the ball will start moving faster and faster as it falls down. By the time it reaches the window 10 meters below, its speed and the time taken can be accurately calculated using our kinematic tools.Understanding free fall is essential for solving problems involving objects falling from heights. It helps in analyzing the motion of the ball and predicting when it will reach specific points, like the window in our problem. It simplifies complex real-world motion into solvable physics problems.