Problem 451 - JEE Physics

Question

A body of mass \(1 \mathrm{~kg}\) is thrown upwards with a velocity \(20 \mathrm{~m} / \mathrm{s}\). It momentarily comes to rest after a height \(18 \mathrm{~m}\). How much energy is lost due to air friction. \((\mathrm{g}=10 \mathrm{~m} / \mathrm{s} 2)\) (A) \(20 \mathrm{~J}\) (B) \(30 \mathrm{~J}\) (C) \(40 \mathrm{~J}\) (D) \(10 \mathrm{~J}\)

Step-by-Step Solution

Answer

The energy lost due to air friction is \(20 \mathrm{~J}\), which corresponds to option (A).

Step 1: Calculate initial kinetic energy (KE)

We can calculate the initial kinetic energy of the body using the formula KE = \(\frac{1}{2}\)mv², where m is the mass of the body (1 kg) and v is its initial velocity (20 m/s). This will give us the initial kinetic energy in Joules (J). KE = \(\frac{1}{2}(1 \mathrm{~kg})(20 \mathrm{~m/s})^2\) = \(200 \mathrm{~J}\)

Step 2: Calculate final potential energy (PE)

Now, we need to find the final potential energy of the body when it reaches the height of 18 m. We can do this using the formula PE = mgh, where m is the mass (1 kg), g is the gravitational acceleration (10 m/s²), and h is the height (18 m). PE = (1 \mathrm{~kg})(10 \mathrm{~m/s²})(18 \mathrm{~m}) = \(180 \mathrm{~J}\)

Step 3: Calculate net energy at the highest point

When the body momentarily comes to rest, its final kinetic energy will be 0 J, as it has stopped completely. Therefore, the net energy at the highest point will be equal to the final potential energy. Net energy = PE = \(180 \mathrm{~J}\)

Step 4: Determine the energy lost due to air friction

To find the energy lost due to air friction, we need to subtract the net energy at the highest point from the initial kinetic energy. Energy loss = Initial KE - Net Energy = \(200 \mathrm{~J} - 180 \mathrm{~J} = 20 \mathrm{~J}\) So, the energy lost due to air friction is \(20 \mathrm{~J}\), which corresponds to option (A).

Key Concepts

Kinetic Energy

Kinetic energy is the energy a body possesses due to its motion. It is dependent on two factors: the mass of the object and its velocity. The formula to calculate kinetic energy is given by:

• KE = \(\frac{1}{2}mv^2\)

where \(m\) stands for the mass of the body, and \(v\) represents its velocity.

In the exercise, a body of mass \(1 \text{ kg}\) is thrown upwards with a velocity of \(20 \text{ m/s}\). Initial kinetic energy can thus be calculated by plugging the values into the formula. This results in a kinetic energy of \(200 \text{ J (Joules)}\).

It's important to understand that as the body moves upwards, it gradually loses kinetic energy due to the opposing force of gravity and air resistance.

Potential Energy

Potential energy is the stored energy of an object due to its position or height. For objects near the Earth's surface, we use the formula:

• PE = \(mgh\)

where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity \( (10 \text{ m/s}^2)\), and \(h\) is the height of the object above the ground.

In this problem, as the body reaches a height of \(18 \text{ m}\), the potential energy becomes \(180 \text{ J}\).

This transformation from kinetic energy to potential energy illustrates that as an object rises, potential energy increases while kinetic energy decreases.

Air Resistance

Air resistance, also known as drag, is the force that opposes the motion of an object through the air. Unlike gravitational force, air resistance depends on the object's velocity, shape, and surface area.

In the context of the exercise, as the body ascends, it not only loses energy due to gravity but also due to air resistance.

• This frictional force results in a loss of mechanical energy.

By the time the object reaches its peak height and potential energy, some of the initial kinetic energy has been lost due to air resistance.

Identifying this energy loss is crucial to understanding how non-conservative forces, like air resistance, affect the total energy balance.

Conservation of Energy

The principle of energy conservation states that energy cannot be created or destroyed in an isolated system. Instead, it transforms from one form to another, like from kinetic to potential energy, while the total mechanical energy remains constant unless acted upon by external forces.

In the given problem, the body starts with a kinetic energy of \(200 \text{ J}\). At its peak, where it stops momentarily, all its kinetic energy has transformed into \(180 \text{ J}\) of potential energy.

• The missing \(20 \text{ J}\) is the energy lost to the air resistance — an external force.

The analysis confirms that even though energy has transformed and some is lost due to external forces, the law holds true overall. The conservation of energy helps in predicting and calculating the changes when multiple forces are at play.