Problem 365 - JEE Physics

Question

A vehicle of \(100 \mathrm{~kg}\) is moving with a velocity of \(5(\mathrm{~m} / \mathrm{s})\). To stop it in \((1 / 10) \mathrm{sec}\), the required force in opposite direction is \(\mathrm{N}\) (A) 50 (B) 500 (C) 5000 (D) 1000

Step-by-Step Solution

Answer

The required force in the opposite direction to stop the vehicle is 5000 N.

Step 1: Identify the given information

We are given the following information: - Mass of the vehicle, m = 100 kg - Initial velocity of the vehicle, v₁ = 5 m/s - Time to stop the vehicle, t = 1/10 s - The vehicle has to stop, which means the final velocity (v₂) will be 0 m/s We are asked to find the required force in the opposite direction (F).

Step 2: Use Newton's second law of motion

According to Newton's second law of motion, the force acting on an object is given by the product of its mass and acceleration, \[F = m \times a\] We need to find the acceleration (a) of the vehicle in order to calculate the force required to stop it.

Step 3: Calculate the acceleration needed to stop the vehicle

To calculate the acceleration needed to stop the vehicle in the given time, we can use the following equation, which relates initial velocity (v₁), final velocity (v₂), acceleration (a), and time (t): \[v₂ = v₁ + at\] Since the vehicle has to stop, we know that the final velocity (v₂) will be 0 m/s. We can now rearrange the equation and solve for the acceleration: \[a = \frac{v₂ - v₁}{t}\]

Step 4: Plug the calculated values into the equation and solve for the force

Now, we will plug in the values of v₁, v₂, and t into the equation to find the acceleration: \[a = \frac{0 - 5}{\frac{1}{10}} = -50 \,m/s^{2}\] The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, which is what we need to stop the vehicle. Now, we can use Newton's second law of motion to find the force: \[F = m \times a\] \[F = 100kg \times -50 \,m/s^{2} = -5000 N \] Since the force is acting in the opposite direction, the magnitude of the force is 5000 N. So, the correct answer is: (C) 5000

Key Concepts

Understanding Acceleration

Acceleration is a key concept in physics, particularly when studying motion. It refers to the rate of change of velocity of an object over time. In essence, it measures how quickly an object speeds up or slows down.

• Acceleration can be positive or negative, depending on whether the object is speeding up or slowing down.
• The units of acceleration are typically meters per second squared (\(m/s^2\)).

In the context of our vehicle problem, knowing the rate at which the vehicle needs to slow down helps us find the force necessary to achieve this. Using the formula:\[a = \frac{v_2 - v_1}{t}\]we can calculate the acceleration by substituting the initial velocity (\(v_1 = 5 \, m/s\)), the final velocity (\(v_2 = 0 \, m/s\)), and the time (\(t = \frac{1}{10} \, s\)). The calculated acceleration tells us not just how fast the velocity changes but also in which direction.

Force Calculation Explained

Force is what causes objects to accelerate. According to Newton's Second Law of Motion, the force acting on an object is the product of its mass and acceleration. This principle can be mathematically expressed using the equation:\[F = m \times a\]

• Force is measured in newtons (N).
• The direction of the force is the same as that of the acceleration.

To calculate the force needed to stop the vehicle in our problem, we substituted the mass (\(m = 100 \, kg\)) and the calculated acceleration (\(a = -50 \, m/s^2\)) into the equation. The negative sign of acceleration indicates the direction of the force is opposite to the direction of the initial motion, which is necessary since we want to stop the vehicle. The resulting force is \(-5000 \, N\), showing that a strong force is needed in the opposite direction to bring the car to a halt.

Connecting to Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It focuses on measurable quantities like displacement, velocity, and acceleration. In this exercise, kinematics were employed to figure out the acceleration required to stop the vehicle.

• Initial and final velocities are fundamental components of kinematic equations.
• Time is a crucial factor that links acceleration to change in velocity in kinematics.

By applying the kinematic equation:\[v_2 = v_1 + at\]we determined how quickly the vehicle should decelerate within a specified timeframe. Solving for acceleration was the first step, followed by using this information to calculate the necessary force to achieve the desired change in motion, connecting back to the principles of Newton’s Second Law.