Problem 59 - Master Resource Book in JEE Main Physics

Question

In a shotput event an athlete throws the shotput of mass \(10 \mathrm{~kg}\) with an inital speed of \(1 \mathrm{~m} \mathrm{~s}^{-1}\) at \(45^{\circ}\) from a height \(1.5 \mathrm{~m}\) above ground. Assuming air resistance to be negligible and acceleration due to gravity to be \(10 \mathrm{~ms}^{-2}\), the kinetic energy of the shotput when it just reaches the ground will be [NCERT Exemplar] (a) \(2.5 \mathrm{~J}\) (b) \(5.0 \mathrm{~J}\) (c) \(52.5 \mathrm{~J}\) (d) \(155.0 \mathrm{~J}\)

Step-by-Step Solution

Answer

Option (d) \(155.0 \mathrm{~J}\).

Step 1: Determine Initial Kinetic Energy

The initial kinetic energy of the shotput when it is thrown is given by the formula \( KE_{initial} = \frac{1}{2}mv^2 \), where \( m \) is the mass of the shotput, and \( v \) is the initial velocity. Substituting the given values, \( m = 10 \mathrm{~kg} \) and \( v = 1 \mathrm{~m/s} \), we get: \[ KE_{initial} = \frac{1}{2} \times 10 \times (1)^2 = 5 \mathrm{~J} \]

Step 2: Calculate Initial Potential Energy

The initial potential energy of the shotput when it is at height is given by the formula \( PE_{initial} = mgh \), where \( m \) is the mass of the shotput, \( g \) is the acceleration due to gravity, and \( h \) is the initial height. Substituting the given values, \( m = 10 \mathrm{~kg} \), \( g = 10 \mathrm{~m/s^2} \), and \( h = 1.5 \mathrm{~m} \), we get: \[ PE_{initial} = 10 \times 10 \times 1.5 = 150 \mathrm{~J} \]

Step 3: Apply Conservation of Energy

The principle of conservation of energy states that the total mechanical energy is conserved when air resistance is negligible. Thus, the sum of initial kinetic and potential energies equals the kinetic energy when the shotput reaches the ground. Therefore, \[ KE_{final} = PE_{initial} + KE_{initial} = 150 \mathrm{~J} + 5 \mathrm{~J} = 155 \mathrm{~J} \]

Step 4: Identify the Correct Option

The computed final kinetic energy when the shotput reaches the ground is \( 155.0 \mathrm{~J} \). This matches option (d).

Key Concepts

Conservation of Energy

The concept of conservation of energy is a fundamental principle in physics. It states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the shotput problem, this means that mechanical energy remains constant as the shotput moves through its trajectory, assuming no air resistance. When the shotput is thrown, its initial energy is a combination of kinetic energy (due to its speed) and potential energy (due to its height above the ground). As it travels, the energy transforms but the total energy remains the same until it reaches the ground.

• Total energy remains constant in a closed system.
• Forms of energy involved: kinetic and potential.
• Total mechanical energy before impact equals total mechanical energy after impact.

Understanding this principle is key to solving many physics problems, particularly those involving motion and energy.
In this exercise, the sum of the kinetic and potential energy at the start is equal to the kinetic energy at the end, when the shotput hits the ground.

Projectile Motion

Projectile motion describes the motion of an object thrown into the air under the influence of gravity, without considering air resistance. The path of a projectile is called a trajectory. In the case of the shotput, the athlete throws the ball at a certain angle and speed, which determines its parabolic path. Key factors influencing projectile motion include the initial speed, angle of projection, and gravitational pull.

• The path followed is a parabola due to gravity.
• Horizontal and vertical motions are independent.
• Initial speed and angle influence the trajectory.

The shotput problem is a classic physics teaching example because it involves predicting where an object will land based on initial conditions. Understanding these concepts is crucial for learning how to calculate time of flight, maximum height reached, and range of the projectile.

Potential Energy

Potential energy is the stored energy of position possessed by an object. In this case, it is gravitational potential energy due to the shotput's height above the ground. The formula for calculating gravitational potential energy is:\[ PE = mgh \]Where \( m \) is the mass, \( g \) is the gravitational acceleration, and \( h \) is the height. Before the shotput is thrown, it has a certain amount of potential energy due to its position. As it falls, this potential energy converts into kinetic energy.

• Depends on height and mass.
• Gravitational potential energy decreases as the shotput falls.
• Energy transformation occurs as the height decreases.

Understanding potential energy is important for analyzing how energy changes forms when objects change their position relative to a gravitational source.

Initial Speed

The initial speed of an object plays a critical role in its motion, especially in projectile motion. It is the speed with which the shotput is thrown horizontally and influences both how far and how high the projectile will go. In this exercise, the initial speed is given as \(1\, ext{m/s}\).

• Initial speed affects distance and height.
• Determines the energy at the start of motion.
• Higher initial speed can result in a longer flight and greater range.

Calculating energy transformations often starts with understanding this initial speed, as it is pivotal to determining the initial kinetic energy.

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the pull of Earth's gravity. It is approximately \(9.81\, ext{m/s}^2\), but for simplicity, this problem uses \(10\, ext{m/s}^2\). This value is crucial for calculating both potential energy and the motion path of the projectile.

• It affects the trajectory and speed of falling objects.
• Used in calculating potential energy and time of flight.
• Assumed constant near Earth's surface.

Understanding gravitational acceleration is vital because it influences both the impact speed of falling objects and the arc described by projectiles. In any physics problem involving falling objects or projectiles, gravity is a key player that affects calculations and outcomes.