# Velocity and Acceleration |Formula,Units and Graph derivation

Velocity is a rate of change in displacement with respect to time. As displacement is a vector quantity having both magnitude and direction, velocity is also a vector quantity.

Acceleration is a rate of change in velocity with respect to time. An object undergoes acceleration whenever an object’s speed increases or decreases (usually referred to as negative acceleration or deceleration or retardation) it changes direction (object moving in a circle is constantly accelerating even if it has constant speed because it is constantly changing its direction).

If a body moves from point A to point B, the total length of the path covered in the distance covered, the shortest possible distance between point A and point B is known as displacement of the object. Distance is a scalar quantity while the displacement is a vector quantity. The distance can only be equal to or greater than displacement.

__Determination of the direction __

Both velocity and acceleration have direction.

The direction of velocity is the direction in which the object moves. It doesn’t matter if the object is speeding up or slowing down. If we say an object has a velocity of 25 m/s due east, then the object is moving in east direction. But if we say that the object is moving with a velocity of -25 m/s due east, then the object is moving in the opposite direction, which is west.

The direction of acceleration is determined by the direction of change in velocity,not by the direction of motion. If an object is speeding up, the direction of acceleration is in the direction of motion, but if the object is slowing down, the direction of acceleration is opposite to the direction of motion.

__Units and dimensions__

Velocity = change in displacement /time

Dimension of displacement = L

Dimension of time = T

Dimension of velocity = [ L T^(-1)]

Unit of velocity = m/s or km/hr

The S. I. unit of velocity is m/s

Acceleration = change in velocity /time

Dimension of velocity = [ L T^(-1)]

Dimension of time = T

Dimension of velocity = [ L T^(-2)]

Unit of velocity = m/s^{2 }or km/hr^{2}

The S. I. unit of acceleration is m/s^{2}

^{ }__Graphs__

*s= displacement *

*t=time*

*v=velocity at a time t*

*u= initial velocity *

*a=acceleration*

On a **displacement-time** graph-

- Slope equals velocity.
- The “y” intercept equals the initial displacement.
- Straight lines imply velocity is constant
- Curved lines imply object is undergoing acceleration or retardation
- Average velocity is given by the slope of the straight line connecting the endpoints of the curve.
- The derivative of a tangent at a point on the curve gives the velocity at that point (instantaneous velocity)
- a positive slope means motion in the positive direction.
- a negative slope means motion in the negative direction.
- zero slope means a state of rest

a = change in velocity /time

a= (v-u) /t

v=u + at

v=ds/dt

On a **velocity-time** graph-

- Slope equals acceleration.
- The “y” intercept equals the initial velocity which is u
- Straight lines imply uniform acceleration.
- Curved lines imply non-uniform acceleration.
- an object undergoing constant acceleration has a straight line graph
- The derivative of the tangent at a point on the curve gives the acceleration at that point
- A positive slope means acceleration
- A negative slope means retardation
- Zero slope means uniform velocity
- The area under the curve gives indicates the displacement of the object

a=dv/dt

In this graph initial velocity v_{o} = u

The area under the v-t curve represents the displacement.

Area under the curve = Area of triangle ABC + Area of rectangle OACD

Displacement =area

s= ½(v-u)t + ut

s= ½(u+at-u)t + ut

s= ut + ½ a t^(2)

On an **acceleration-time** graph-

- The “y” intercept equals the initial acceleration.
- an object undergoing constant acceleration has a horizontal line with zero slopes on the graph
- The area under the curve gives the velocity of the object

__Derivation of formula __

a=dv/dt

Or, a dt= dv

Integrating both sides, where time is from t=0 to t=t and velocity is from v=u to v=v

at =v – u

**v = u + at**

v=ds/dt

Or, v dt= ds

Or, (u + at)dt =ds

Integrating both sides, where time is from t=0 to t=t and displacement is from s=0(Let initial displacement =0) to s=s

**s=ut + ½ a t^(2) **

a= dv/dt

Or, a= (dv/ds)(ds/dt)

Or, a = v dv/ds

Or, a ds=vdv

Integrating both sides, where displacement is from s=0(Let initial displacement =0) to s=s and velocity is from v=u to v=v

as= ½ (v^2 -u^2)

**v^2 =u^2 + 2as**

Thus the following three formulae are the three equations of motion

- v = u + at
- s=ut + ½ a t^(2)
- v^2 =u^2 + 2as