Motion Notes

Introduction

Motion is a fundamental concept in physics that describes the change in position of an object over time. In the context of JEE Advanced Physics, understanding motion involves grasping several key principles and equations that govern how objects move. This study note will break down the topic of motion into manageable sections, covering everything from basic definitions to complex equations and applications.

Types of Motion

1. Translational Motion

Translational motion occurs when an object moves along a path in any of the three dimensions. This type of motion can be further divided into:

• Rectilinear Motion: Motion along a straight line.
• Curvilinear Motion: Motion along a curved path.

2. Rotational Motion

Rotational motion involves an object rotating about an axis. Each point in the object follows a circular path around the axis of rotation.

3. Oscillatory Motion

Oscillatory motion is repetitive back-and-forth movement about a central point, such as the motion of a pendulum.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion.

1. Displacement, Velocity, and Acceleration

• Displacement ($\vec{s}$): A vector quantity that denotes the change in position of an object. It has both magnitude and direction. $$\vec{s} = \vec{x}_f - \vec{x}_i$$ where $\vec{x}_f$ is the final position and $\vec{x}_i$ is the initial position.
• Velocity ($\vec{v}$): The rate of change of displacement with respect to time. It is also a vector quantity. $$\vec{v} = \frac{d\vec{s}}{dt}$$
• Acceleration ($\vec{a}$): The rate of change of velocity with respect to time. $$\vec{a} = \frac{d\vec{v}}{dt}$$

2. Equations of Motion for Uniform Acceleration

In cases of uniform acceleration, the following equations of motion can be used:

1. $v = u + at$
2. $s = ut + \frac{1}{2}at^2$
3. $v^2 = u^2 + 2as$

where:

• $u$ is the initial velocity,
• $v$ is the final velocity,
• $a$ is the acceleration,
• $t$ is the time,
• $s$ is the displacement.

3. Projectile Motion

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path under the influence of gravity alone.

Key Equations

• Horizontal Range: $R = \frac{u^2 \sin 2\theta}{g}$
• Maximum Height: $H = \frac{u^2 \sin^2 \theta}{2g}$
• Time of Flight: $T = \frac{2u \sin \theta}{g}$

Solution

Given:

• $u = 20$ m/s
• $\theta = 30^\circ$
• $g = 9.8$ m/s²

Time of Flight: $$T = \frac{2u \sin \theta}{g} = \frac{2 \times 20 \times \sin 30^\circ}{9.8} = \frac{20}{9.8} \approx 2.04 \text{ s}$$

Maximum Height: $$H = \frac{u^2 \sin^2 \theta}{2g} = \frac{20^2 \times (\sin 30^\circ)^2}{2 \times 9.8} = \frac{400 \times 0.25}{19.6} \approx 5.10 \text{ m}$$

Horizontal Range: $$R = \frac{u^2 \sin 2\theta}{g} = \frac{20^2 \times \sin 60^\circ}{9.8} = \frac{400 \times \sqrt{3}/2}{9.8} \approx 35.36 \text{ m}$$

Dynamics

Dynamics is the study of the forces and torques that cause motion.

1. Newton's Laws of Motion

• First Law (Law of Inertia): An object will remain at rest or in uniform motion unless acted upon by a net external force.

• Second Law: The rate of change of momentum of an object is directly proportional to the net external force acting on it and occurs in the direction of the force. $$\vec{F} = m\vec{a}$$
• Third Law: For every action, there is an equal and opposite reaction.

2. Work, Energy, and Power

• Work: Work is done when a force causes displacement. $$W = \vec{F} \cdot \vec{s} = F s \cos \theta$$
• Kinetic Energy: The energy possessed by an object due to its motion. $$KE = \frac{1}{2}mv^2$$
• Potential Energy: The energy possessed by an object due to its position or configuration. $$PE = mgh$$
• Power: The rate at which work is done. $$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$$

Circular Motion

1. Uniform Circular Motion

In uniform circular motion, an object moves in a circle with constant speed. Although the speed is constant, the velocity is not due to the continuous change in direction.

• Centripetal Acceleration: The acceleration directed towards the center of the circle. $$a_c = \frac{v^2}{r}$$
• Centripetal Force: The net force causing centripetal acceleration. $$F_c = m a_c = m \frac{v^2}{r}$$

2. Non-Uniform Circular Motion

In non-uniform circular motion, the speed of the object changes as it moves along the circular path. This involves both tangential acceleration ($a_t$) and centripetal acceleration ($a_c$).

Relative Motion

Relative motion is the calculation of the motion of an object with respect to some other moving object.

1. Relative Velocity

If two objects A and B are moving with velocities $\vec{v}_A$ and $\vec{v}_B$ respectively, the velocity of A relative to B is given by:

$$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$

Solution

Given:

• $\vec{v}_A = 60$ km/h
• $\vec{v}_B = 80$ km/h

Relative velocity: $$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B = 60 - 80 = -20 \text{ km/h}$$

The negative sign indicates that the first car is moving slower than the second car by 20 km/h.

Conclusion

Understanding motion is crucial for solving problems in JEE Advanced Physics. By mastering the concepts of kinematics, dynamics, and relative motion, you will be well-equipped to tackle a wide range of questions. Remember to practice consistently and pay attention to the details, such as units and vector directions, to avoid common mistakes.