Complete Kinematics Summary

Motion, Velocity, Acceleration, Projectile Motion, and Circular Motion

Basic Concepts
Types of Motion
Kinematics Equations
Projectile Motion
Circular Motion

1. Basic Kinematics Concepts

1.1 Displacement

The displacement of a particle is defined as the difference between its final position and its initial position. We represent the displacement as \(\Delta x\).

\[\Delta x = x_f - x_i\]

The subscripts \(i\) and \(f\) refer to the initial and final positions.

1.2 Average Velocity and Average Speed

The average velocity of an object travelling along the \(x-\)axis is defined as the ratio of its displacement to the time taken for that displacement.

\[v_{av} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}\]

1.3 Average Acceleration

Average acceleration is defined as the ratio of the change in velocity to the time taken.

\[a_{av} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}\]

1.4 Instantaneous Velocity

The magnitude of instantaneous velocity at a given instant is called instantaneous velocity at that instant.

\[v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}\]

The instantaneous velocity function is the derivative with respect to the time of the displacement function.

\[v(t) = \frac{dx(t)}{dt}\]

1.5 Instantaneous Acceleration

Instantaneous acceleration is defined analogous to the method for defining instantaneous velocity. That is, instantaneous acceleration is the value approached by the average acceleration as the time interval for the measurement becomes closer and closer to zero.

The Instantaneous acceleration function is the derivative with respect to time of the velocity function.

\[a(t) = \frac{dv(t)}{dt}\]

2. Types of Motion

1. At Rest
Quantity Value
Displacement (\(x\)) \(c\) (constant)
Velocity (\(v\)) \(0\)
Acceleration (\(a\)) \(0\)
2. Motion with Constant Velocity
Quantity Value
Displacement (\(x\)) \(x = v_0 t + x_0\)
Velocity (\(v\)) \(v_0\) (constant)
Acceleration (\(a\)) \(0\)
3. Motion with Constant Acceleration
Quantity Value
Displacement (\(x\)) \(x = x_0 + v_0 t + \frac{1}{2} a_0 t^2\)
Velocity (\(v\)) \(v = v_0 + a_0 t\)
Acceleration (\(a\)) \(a_0\) (constant)
4. Motion with Constant Deceleration
Quantity Value
Displacement (\(x\)) \(x = x_0 + v_0 t - \frac{1}{2} a_0 t^2\)
Velocity (\(v\)) \(v = v_0 - a_0 t\)
Acceleration (\(a\)) \(-a_0\) (constant)

3. Motion with Constant Acceleration

The equations of kinematics for motion with constant acceleration are summarized as:

\[v = v_0 + at\]
\[x = x_0 + v_0 t + \frac{1}{2} at^2\]
\[v^2 = v_0^2 + 2a(x - x_0)\]
\[x = x_0 + \frac{1}{2}(v_0 + v)t\]

Where,

  • \( x_0 \) = Initial position coordinate
  • \( x \) = Final position coordinate
  • \( v_0 \) = Initial velocity
  • \( v \) = Final velocity
  • \( a \) = Acceleration (constant)
  • \( t \) = Elapsed time

3.1 Free Fall

Motion that occurs solely under the influence of gravity is called free fall.

The equations of kinematics may be modified for free fall as:

\[v = v_0 - gt\]
\[y = y_0 + \frac{1}{2}(v_0 + v)t\]
\[y = y_0 + v_0 t - \frac{1}{2}gt^2\]
\[v^2 = v_0^2 - 2g(y - y_0)\]

Note:

The signs of \( v \) and \( v_0 \) are determined by their directions relative to the chosen \( +y \) axis.

3.2 Two-Dimensional Motion

For two-dimensional motion in the plane, the \( x \) and \( y \) components of these equations are:

X-Components:

\[v_x = v_{0x} + a_x t\]
\[x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2\]
\[x = x_0 + \frac{1}{2} (v_{0x} + v_x) t\]
\[v_x^2 = v_{0x}^2 + 2a_x (x - x_0)\]

Y-Components:

\[v_y = v_{0y} + a_y t\]
\[y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2\]
\[y = y_0 + \frac{1}{2} (v_{0y} + v_y) t\]
\[v_y^2 = v_{0y}^2 + 2a_y (y - y_0)\]

4. Projectile Motion

A projectile motion near the surface of the earth consists of two independent motions, a horizontal motion at constant speed and a vertical one subject to the acceleration due to gravity.

4.1 Equations of Kinematics for Projectile Motion

\[x = v_{0x} t \quad \text{(Horizontal motion)}\]
\[v_y = v_{0y} - gt \quad \text{(Vertical velocity)}\]
\[y = y_0 + v_{0y} t - \frac{1}{2}gt^2 \quad \text{(Vertical position)}\]
\[v_y^2 = v_{0y}^2 - 2g(y - y_0) \quad \text{(Vertical motion equation)}\]

4.2 Important Projectile Motion Parameters

1. Time of Flight

\[T = \frac{2v_0 \sin \theta}{g}\]

2. Horizontal Range

\[R = \frac{v_0^2 \sin 2\theta}{g}\]

3. Maximum Height

\[H = \frac{v_0^2 \sin^2 \theta}{2g}\]

4. Trajectory Equation

\[y = x \tan \theta - \frac{1}{2} \frac{g}{(v_0 \cos \theta)^2} x^2\]

The trajectory of a projectile is a parabola.

5. Circular Motion

Let us consider a particle which moves in a circular path of radius \( r \) and constant speed \( v \).

5.1 Angular Speed

\[\omega = \frac{v}{r}\]

5.2 Centripetal Acceleration

\[a = \frac{\Delta v}{\Delta t} = \frac{v^2}{r}\]

5.3 Tangential and Normal Components of Acceleration

Key Points:

  1. The velocity vector is always tangential to the path.
  2. The acceleration vector may have two components: one tangential to the path and one perpendicular to the path.
    • The component of the acceleration parallel to the path is due to a change in speed. When the speed is increasing, the tangential component \( a_t \) points in the same direction as the velocity; when the speed is decreasing, the tangential component points opposite to the velocity.
    • When the path of an object curves, there is a component of the acceleration perpendicular to the velocity. This component of the acceleration \( a_c \) points toward the inside of the curve.
  3. The total acceleration is the vector sum of the tangential and centripetal components.