Motion In A Straight Line / Plane

Table of Contents

Basic Kinematic Quantities

Quantity	Formula	Key Point
Displacement	\(\displaystyle\Delta x=x_f-x_i\)	Shortest directed distance
Average Velocity	\(\displaystyle\frac{\text{Displacement}}{\text{Time}}\)	Depends on displacement
Average Speed	\(\displaystyle\frac{\text{Total Distance}}{\text{Total Time}}\)	Depends on actual path
Instantaneous Velocity	\(\displaystyle v=\frac{dx}{dt}\)	Velocity at an instant
Acceleration	\(\displaystyle a=\frac{dv}{dt}\)	Rate of change of velocity
Variable acceleration	\(\displaystyle a=v\frac{dv}{dx}\)	Useful when acceleration depends on distance

2. Average Speed Shortcuts

Condition	Formula
Equal distances covered at \(\displaystyle v_1,v_2\)	\(\displaystyle\bar v=\frac{2v_1v_2}{v_1+v_2}\)
Equal time intervals	\(\displaystyle\bar v=\frac{v_1+v_2+…}{n}\)
General case	\(\displaystyle\bar v=\frac{x_1+x_2+…}{x_1/v_1+x_2/v_2+…}\)

Use only when acceleration is constant.

Distance in n^th second:

Take upward direction positive:

Acceleration:

Maximum height:

Time of ascent:

Total time of flight:

Velocity after falling through height

Straight Line

Same direction:

Opposite direction:

Plane Motion

Relative velocity:

Relative acceleration:

\(\displaystyle\vec a_{AB}=\vec a_A-\vec a_B\)

For angle

Direction:

Motion in x and y directions are independent.

Horizontal motion:

Vertical motion:

Velocity components:

Projectile Motion Formulas

Quantity	Formula
Time of flight	\(\displaystyle\frac{2u\sin\theta}{g}\)
Maximum height	\(\displaystyle\frac{u^2\sin^2\theta}{2g}\)
Horizontal range	\(\displaystyle\frac{u^2\sin2\theta}{g}\)

Maximum range: