Problem 72 - A Complete Resource Book in Physics for JEE Main

Question

A body with mass \(5 \mathrm{~kg}\) is acted upon by a force \(F=(-3 \hat{i}+4 \hat{j}) N .\) If its initial velocity at \(t=0\) is \(v=(6 \hat{i}-12 \hat{j}) \mathrm{ms}^{-1}\), the time at which it will just have a velocity along the \(y\)-axis is \(\begin{array}{lll}\text { (A) Never } & \text { (B) } 10 \mathrm{~s} & \text { (C) } 2 \mathrm{~s}\end{array}\) (D) \(15 \mathrm{~s}\)

Step-by-Step Solution

Answer

In short, to find the time at which the body with mass \(5\,\text{kg}\) and force \(F=(-3\hat{i}+4\hat{j}) \,\text{N}\) has velocity only along the y-axis, we first calculate the acceleration and then integrate it to find the velocity function. By setting the x-component of the velocity function to zero and solving for time, we obtain \(t = 10\,s\), which is the time at which the body has a velocity only along the y-axis. Hence, the correct answer is (B) 10 s.

Step 1: 1. Calculate acceleration

To find the acceleration, we need to divide the force vector by the mass of the body. Since the force is given as \(F = (-3\hat{i} + 4\hat{j}) \,\text{N}\) and mass is \(m = 5\,\text{kg}\), the acceleration vector can be calculated as follows: \(a = \frac{F}{m} = \frac{-3\hat{i} + 4\hat{j}}{5}\)

Step 2: 2. Integrate the acceleration function to find the velocity function

Integrate the acceleration function with respect to time t to find the velocity function. Remember that the initial velocity at \(t=0\) is given as \(v(0)=(6 \hat{i}-12 \hat{j})\,\text{ms}^{-1}\). \(v(t) = \int a dt = \int \frac{-3\hat{i} + 4\hat{j}}{5} dt\) \(v(t) = -\frac{3}{5}t\hat{i} + \frac{4}{5}t\hat{j} + v(0)\) Now substitute the initial velocity, \(v(0) = 6\hat{i} - 12\hat{j}\), into the equation: \(v(t) = -\frac{3}{5}t\hat{i} + \frac{4}{5}t\hat{j} + 6\hat{i} - 12\hat{j}\)

Step 3: 3. Find the time at which the velocity is only along the y-axis

We need to find the time at which the x-component of the velocity will be zero, while the y-component has a non-zero value. To do this, we first make the x-component equal to zero and solve for the time: \(-\frac{3}{5}t + 6 = 0\) Now, solve for t: \(t = \frac{6}{3/5} = 10\,s\) Now, let's see if the y-component of the velocity at the same time is non-zero: \(v_y(t) = \frac{4}{5}(10) - 12 = 4\) Since at \(t = 10\,s\), the y-component of the velocity is 4 m/s, it means that the body has a velocity only along the y-axis at this time. Thus, the correct answer is (B) 10 s.

Key Concepts

Force and Motion

In physics, force and motion are central concepts that describe how objects move and interact with each other. A force is any interaction that, when unopposed, will change the motion of an object. This can include things like pushing, pulling, gravity, or even magnetic attractions. Newton's Second Law of Motion is the key principle that links force and motion together, expressed as \( F = ma \), where

• \( F \) is the force applied to the object,
• \( m \) is the mass of the object, and
• \( a \) is the acceleration of the object.

This formula tells us that the acceleration of an object is directly proportional to the force acting upon it and inversely proportional to its mass. Thus, the heavier an object, the more force is needed to accelerate it. In the original exercise, this principle is applied by dividing the force vector by the object's mass to find the acceleration vector.

Vectors in Physics

Vectors are essential tools in physics used to represent quantities that have both a magnitude and a direction. Force, velocity, and acceleration are all vector quantities. A vector can be represented graphically by an arrow;

• the length of the arrow indicates the magnitude of the vector, and
• the direction of the arrow shows the direction of the vector.

In mathematical terms, vectors are expressed in terms of their components along coordinate axes, typically denoted as \( \hat{i} \) and \( \hat{j} \) for the x and y components, respectively. For example, the force \( F = (-3\hat{i} + 4\hat{j}) \,\text{N} \) in the original exercise implies that

• there is a force of \(-3 \,\text{N} \) in the x-direction, and
• \(4 \,\text{N} \) in the y-direction.

Understanding vectors is crucial for physics problem-solving, especially when analyzing motion in two or more dimensions.

Velocity and Acceleration

Velocity is a vector quantity that represents the rate of change of an object's position. It tells you both how fast something is moving and in which direction. On the other hand, acceleration is the rate of change of velocity and is also a vector quantity representing how the velocity of an object is changing over time. An object accelerates if its speed changes, its direction changes, or both.
In the exercise, the acceleration vector is determined by dividing the force by the mass. This involves treating force and mass as vectors, resulting in an acceleration expressed in terms of the unit vectors \( \hat{i} \) and \( \hat{j} \). Once the acceleration is known, it can be integrated over time to find the velocity, incorporating any initial velocity present. Recognizing how velocity and acceleration interplay allows one to understand the dynamics of motion for predictive and analytical purposes.

Integration in Physics

Integration is a fundamental mathematical tool used in physics to find quantities based on their rates of change. For motion problems, integration is used to go from acceleration to velocity and from velocity to position.

• When you integrate acceleration (a function of time), you get velocity.
• When you integrate velocity, you get the position of the object as a function of time.

The original step-by-step solution shows this application by integrating the acceleration vector over time to find the velocity vector. This is done while considering initial conditions, such as the initial velocity.
The function \( v(t) = \int a(t) dt \) allows incorporation of both constant and variable forces that act over time, providing a comprehensive description of motion. This is key in fully analyzing scenarios involving force and motion, offering a deeper understanding of how an object’s velocity changes with respect to time.