Question

Assertion and Reason are given in following questions. Each question have four option. One of them is correct it. (1) If both assertion and reason and the reason is the correct explanation of the Assertion. (2) If both assertion and reason are true but reason is not the correct explanation of the assertion. (3) If the assertion is true but reason is false. (4) If the assertion and reason both are false. Assertion: stopping distance \(=[\\{\) Kinetic energy \(\\} /\\{\) Stopping force \(\\}]\) Reason: Work done in stopping a body is equal to K.E. of the body. (A) 1 (B) 2 (C) 3 (D) 4

Short answer

(A) 1

Step-by-step solution

Understand the Assertion

The Assertion states that the stopping distance is equal to the kinetic energy divided by the stopping force. Stopping distance is the distance required for an object to come to a complete stop. In general, the stopping distance depends on several factors such as the initial speed, the mass of the object, and the stopping force. The most common formula for stopping distance is given by: \(d = \frac{v^2}{2a}\) where: \(d\) = stopping distance, \(v\) = initial velocity, \(a\) = deceleration or stopping force/mass. Now, since kinetic energy is given by the formula: K.E. = \(\frac{1}{2}mv^2\) We need to check if the stopping distance can be represented as the kinetic energy divided by the stopping force.

Understand the Reason

The Reason states that the work done in stopping a body is equal to its kinetic energy. According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. W = ΔK.E. Since the final kinetic energy of a stopped object is 0, we can say that the work done in stopping a body is equal to its initial kinetic energy: W = K.E. As we know that work done (W) is also given by the formula: W = F × d Where F is the stopping force and d is the distance.

Determine the relationship between Assertion and Reason

Now, let's relate the stopping distance formula with the work-energy theorem. From the work-energy theorem, we have: \(W = F \times d = K.E.\) We can express the stopping distance as: \(d = \frac{K.E.}{F}\) The stopping distance formula derived here matches with the Assertion. This demonstrates that the Assertion is true. Additionally, the Reason (work done in stopping a body is equal to its kinetic energy) is also true and provides the correct explanation for this Assertion.

Choose the correct answer

Since both the Assertion and Reason are true, and the Reason provides the correct explanation for the Assertion, the correct answer is (A) 1.

Key concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics. It is the energy that an object possesses due to its motion. Any moving object, whether it be a speeding car or a rolling ball, has kinetic energy. The formula for kinetic energy is given by:\[K.E. = \frac{1}{2}mv^2\]Where:

• \(K.E.\) is the kinetic energy
• \(m\) is the mass of the object
• \(v\) is the velocity of the object

This energy depends on two factors: the mass of the object and its speed. An increase in speed results in a significant increase in kinetic energy because velocity is squared in the formula. For instance, if you double the speed of an object, its kinetic energy increases by four times.
Understanding kinetic energy is essential as it helps in analyzing the behavior of moving objects. It also plays a crucial role in determining how much work is required to stop a moving object. The kinetic energy must be converted into other forms of energy to bring the object to a halt.

Work-Energy Theorem

The work-energy theorem is a principle that connects the concepts of work and kinetic energy. It states that the work done by the net force acting on an object is equal to the change in its kinetic energy.Mathematically, the work-energy theorem is expressed as:\[W = \Delta K.E.\]Here:

• \(W\) is the work done on the object
• \(\Delta K.E.\) is the change in kinetic energy

This theorem implies that if you perform work on an object, such as applying a force to move it, this work results in a change in the object's kinetic energy. For example, if a car comes to a stop, the work done by the brakes reduces the car's kinetic energy to zero.
The work-energy theorem is a powerful tool in physics. It allows us to solve problems where the force varies or when the details of motion are complex. By focusing on the initial and final energies, it simplifies our calculations.

Stopping Distance

Stopping distance is the distance over which a moving object comes to a complete stop. It is a crucial factor in road safety and vehicle design. The stopping distance is affected by:

• The initial velocity of the object
• The mass of the object
• The magnitude of the stopping force

The standard formula for stopping distance based on deceleration is:\[d = \frac{v^2}{2a}\]Where:

• \(d\) is the stopping distance
• \(v\) is the initial velocity
• \(a\) is the deceleration or \(F/m\)

If we incorporate the definition of kinetic energy and the work-energy theorem, we can express the stopping distance as:\[d = \frac{K.E.}{F}\]In practical terms, knowing the stopping distance helps in assessing how much space a vehicle requires to stop safely. It is crucial for designing safe speeds and distances between vehicles on the road.

Stopping Force

Stopping force is the force applied to bring a moving object to a stop. This force is vital in situations like braking in vehicles. It is applied in the opposite direction to the motion, causing the object to decelerate and eventually stop. The relation between stopping force and stopping distance is given by:\[W = F \times d = K.E.\]Where:

• \(F\) is the stopping force
• \(d\) is the stopping distance

The formula indicates that the work done, equivalent to the kinetic energy, depends on both force and distance. To minimize stopping distance, one can increase the stopping force. However, this can be challenging in real-world situations, like on slippery roads where friction is reduced.
Understanding stopping force aids in optimizing braking systems and enhancing safety features in vehicles. It is an essential factor in ensuring that vehicles can stop efficiently and safely in varied driving conditions.