# kinematics Definition & Equations

 Definitions

Displacement is the distance travelled along a specified direction.

Speed is the rate of change of distance.

Velocity is the rate of change of displacement.

Acceleration is the rate of change of velocity.

 Graphical representation of motion

Displacement-time graph:
Gradient of s-t graph = Instantaneous velocity v = ds/dt

Velocity-time graph:
Area under v-t graph = displacement at time t
Gradient of v-t graph = Instantaneous acceleration a = dv/dt

Acceleration-time graph:
Area under a-t graph = change in velocity at time t

The graphs representing a uniformly acceleration motion are as follows:

 Equations for uniform accelerated motion

v = u + a t
s = <v>t = ½(u + v) t
v2 = u2+ 2a s
s = u t + ½at2

 Projectile motion

A ball is thrown in air at an angle to the ground. Neglecting air resistance, its velocity vector v is changing but the acceleration due to gravity g is constant. The horizontal component of v is constant since there is no horizontal acceleration. But the vertical component of v is changing due to g.

In projectile motion, the horizontal and the vertical motion are treated independently.

For the horizontal motion, the particle is undergoing uniform motion, i.e. its horizontal component of velocity remains constant.
Thus sx = uxt
where uxis the horizontal velocity component and sx is the horizontal displacement (range)

For the vertical motion: the particle undergoing uniform accelerated motion due to g.

Thus sy= uy t + ½ a t2

vy= uy + a t

vy 2= uy2 + 2 a sy

where sy is the vertical displacement, uy is the initial vertical velocity component, and vy is the final vertical velocity component.

On solving the equations for horizontal and vertical motion, it can be deduced that the
Maximum Height H = u2sin2θ/2g
Range R = u2sin 2θ/g