Problem 73 - A Complete Resource Book in Physics for JEE Main

Question

A car of mass \(m\) starts from rest and acquires a velocity along east, \(v=v \hat{i}(v>0)\) in two seconds. Assuming the car moves with uniform acceleration, the force exerted on the car is (A) \(\frac{m v}{2}\) eastward and is exerted by the car engine. (B) \(\frac{m v}{2}\) eastward and is due to the friction on the tyres exerted by the road. (C) more than \(\frac{m v}{2}\) eastward exerted due to the engine and overcomes the friction of the road. (D) \(\frac{m v}{\text { exerted by the engine. }}\)

Step-by-Step Solution

Answer

The correct answer to the problem is (A). By using the first equation of motion, we calculated the acceleration as \(a = \frac{v}{2}\). Then, applying Newton's second law (F = ma), we found the force to be \(F = m \times \frac{v}{2}\) eastward. This force is exerted primarily from the car engine, making option (A) the correct choice. Options (B) and (C) were incorrect as the force is not due to friction or greater than \(\frac{m v}{2}\). Option (D) was incomplete.

Step 1: Identify known values

First, we have the following known values: - Mass of the car: \(m\) - The initial velocity of the car: \(u = 0\) - The final velocity of the car: \(v = v\hat{i}\) (eastward) - The time: \(t = 2s\)

Step 2: Understand the concept of uniform acceleration

Uniform acceleration means that the car accelerates at a constant rate during its motion. We will be using the first equation of motion to find the acceleration, which is given by: \(v = u + at\)

Step 3: Calculate the acceleration

In this case, since the initial velocity (\(u\)) is 0, we can solve for the acceleration: \(v = at\) \(a = \frac{v}{t}\) We get the acceleration to be: \(a = \frac{v}{2}\)

Step 4: Use Newton's second law of motion to find the force exerted

Newton's second law of motion states that force is equal to the mass multiplied by the acceleration (\(F = ma\)). Therefore, using the calculated acceleration and the given mass (\(m\)), we can find the force exerted on the car: \(F = m \times a\) \(F = m \times \frac{v}{2}\)

Step 5: Understand the origin of the force

The force exerted on the car is primarily due to its engine. However, the friction force between the tires and the road is also an important contribution to the overall force exerted. Furthermore, this force is eastward as the car is moving eastward. Now, let's check for the answer in the given options. (A) The force is indeed \(\frac{mv}{2}\) eastward due to the engine, so this option is correct. (B) The force is of the right magnitude but is exerted by the car engine, not friction. So, this option is incorrect. (C) The force is not more than \(\frac{mv}{2}\) eastward, so this option is also incorrect. (D) This option is not complete, and it seems there's some text missing. Therefore, it cannot be considered as a valid answer. Hence, the correct answer is: (A) \(\frac{mv}{2}\) eastward and is exerted by the car engine.

Key Concepts

Uniform Acceleration

Uniform acceleration is a fundamental concept in JEE Main Physics that describes an object speeding up or slowing down at a constant rate. This means that the object's velocity changes by the same amount in every equal time period. In the context of the textbook exercise, a car is accelerating uniformly, indicating that the force applied by the engine is constant, and thus, the acceleration is consistent.

However, understanding uniform acceleration requires not only recognizing that the acceleration (\( a \) is constant but also knowing how to apply this knowledge to real-world problems, such as the scenario presented in the exercise. This is where the equations of motion come into play, providing a mathematical way to relate acceleration to velocity and time, which is expressed in the formula: \( v = u + at \), where \( u \) is the initial velocity, \( v \) is the final velocity, and \( t \) is the time. This formula is crucial in calculating the acceleration and the subsequent force exerted on the car.

Newton's Second Law of Motion

Newton's Second Law of Motion is a cornerstone of physics and is especially critical in solving problems for JEE Main Physics. Stated simply, the law asserts that the force (\( F \)) acting on an object is equal to the mass (\( m \)) of the object multiplied by the acceleration (\( a \) caused by the force. Expressed mathematically, it is \( F = ma \).

In the application of Newton's Second Law to the textbook exercise, we determine that the force propelling the car forward comes from the car engine, embodying the principle that forces result in acceleration. Breaking down the law further, not only does it define the relationship between force, mass, and acceleration, but it also implies that force has both magnitude and direction, which aligns with the fact that the car moves eastward and the force is described accordingly.

Equations of Motion

The Equations of Motion serve as the mathematical backbone for describing the motion of objects under the influence of forces. In JEE Main Physics, these equations allow us to predict the future position and velocity of objects undergoing uniform acceleration. The problem from the textbook illustrates the use of one of these equations (\( v = u + at \) to calculate the velocity of the car based on its acceleration.

The three main equations of motion are:

• 1st Equation: \( v = u + at \)
• 2nd Equation: \( s = ut + \frac{1}{2}at^2 \)
• 3rd Equation: \( v^2 = u^2 + 2as \)

In the given scenario, the first equation is applied since we have initial velocity (\( u = 0 \) , final velocity (\( v \)), and time (\( t \) known to us, which contributes to a clear understanding of how uniform acceleration and forces explain the motion of the car. Even though the exercise only requires the first equation, familiarity with all three is essential for mastering the topics in JEE Main Physics.