Problem 69 - Master Resource Book in JEE Main Physics

Question

A stone is dropped from the top of a tall tower. The ratio of the kinetic energy of the stone at the end of three seconds to the increase in the kinetic energy of the stone during the next three seconds is (a) \(1: 1\) (b) \(1: 2\) (c) \(1: 3\) (d) \(1: 9\)

Step-by-Step Solution

Answer

The ratio is \(1:3\) (option c).

Step 1: Understanding the Problem

The problem involves a stone falling freely under gravity. We need to find the ratio of kinetic energy (KE) at the end of 3 seconds to the increase in KE over the next 3 seconds. This requires using the kinetic energy formula for freely falling objects.

Step 2: Formulating the Equations

The formula for kinetic energy is "\( KE = \frac{1}{2} mv^2 \)". Here, "\( m \)" is mass and "\( v \)" is velocity. Since the stone is dropped from rest, "\( v = gt \)" where "\( g \)" is the acceleration due to gravity (approximately \( 9.8 \ m/s^2 \)) and "\( t \)" is time in seconds.

Step 3: Calculating Kinetic Energy at 3 Seconds

At \( t = 3 \) seconds, velocity "\( v_1 = g \times 3 \)". Substitute this into the kinetic energy formula: \( KE_{3s} = \frac{1}{2} m (g \times 3)^2 = \frac{1}{2} mg^2 \times 9 \).

Step 4: Calculating Velocity at 6 Seconds

At \( t = 6 \) seconds, the new velocity "\( v_2 = g \times 6 \)". Insert this into the kinetic energy formula: \( KE_{6s} = \frac{1}{2} m (g \times 6)^2 = \frac{1}{2} mg^2 \times 36 \).

Step 5: Finding Increase in Kinetic Energy Over Next 3 Seconds

The increase in kinetic energy from 3 to 6 seconds is \( KE_{6s} - KE_{3s} \). Using previous results: \( KE = \frac{1}{2} mg^2 \times 36 - \frac{1}{2} mg^2 \times 9 = \frac{1}{2} mg^2 \times 27 \).

Step 6: Calculating the Ratio

The ratio of kinetic energy at 3 seconds to the increase over the next 3 seconds is \( \frac{\frac{1}{2} mg^2 \times 9}{\frac{1}{2} mg^2 \times 27} = \frac{9}{27} = \frac{1}{3} \). Thus, the ratio is \( 1:3 \).

Key Concepts

Free Fall Motion

Free fall motion is a fascinating concept where an object moves under the influence of gravity alone, without being propelled by any other forces such as air resistance or friction. Imagine dropping a stone from the top of a tower; its motion is entirely dictated by the gravitational pull of the Earth.

In real-life scenarios, we often encounter free fall when objects are dropped or when someone jumps off a diving board. Here, the only significant force acting on the object is gravity, which causes the object to accelerate downward.

A few key characteristics of free fall motion include:

• Uniform acceleration: All objects experience the same gravitational pull and accelerate at the same rate when in free fall.
• Velocity increases over time: As an object falls, its speed continually rises, leading to drastic changes in its kinetic energy.
• Initial velocity: In many problems, such as this one, a stone is dropped from rest. This means its initial velocity is zero.

Free fall is essential for understanding how gravity affects motion and plays a crucial role in calculating the energy an object experiences during descent.

Acceleration Due to Gravity

In the context of free fall, the acceleration due to gravity, denoted as "\( g \)", is a key factor influencing how objects move. On Earth, this acceleration is approximately \( 9.8 \, \text{m/s}^2 \). This value represents how quickly an object's velocity increases as it falls.

Characteristics of gravitational acceleration include:

• Uniform for all objects: Independent of mass or composition, all particles or bodies near the surface of the Earth fall with this same acceleration when air resistance is negligible.
• Influence on time and speed: Since \( g \) directly affects the velocity equation \( v = gt \), it helps determine how fast an object will travel as time progresses.
• Crucial for calculations: In exercises like this, \( g \) is essential to compute the velocity at different time points, which then feeds into calculating kinetic energy.

A sound understanding of this constant accelerative force helps in solving many physics problems that involve motion and energy.

Kinetic Energy Formula

The kinetic energy of an object in motion is captured by the formula: \( KE = \frac{1}{2} mv^2 \). Here, \( m \) represents the object's mass, and \( v \) denotes its velocity. Kinetic energy measures the energy something possesses due to its motion.

When dealing with free fall, as in the exercise, velocity is directly influenced by gravity. This means that as an object like a stone falls and speeds up due to gravitational forces, its kinetic energy grows significantly.

Important points about kinetic energy include:

• Scalar quantity: Kinetic energy does not have a direction, only magnitude.
• Proportional to the square of velocity: This means doubling the velocity increases kinetic energy by a factor of four.
• Derived from motion: In free fall, the transformation of potential energy (height-based) to kinetic energy (motion-based) illustrates energy conservation principles.

In physics problems, understanding how to manipulate the kinetic energy formula and apply it to varying scenarios like changing times or heights is crucial for accurate problem solving.