Motion in 1-D Fundamentals - Acceleration

Description:

Definition

Rate of change of velocity w.r.t. to time is called Acceleration.

Average Acceleration = Change in Velocity/Total time = Δv→/Δt = v_f - v_i/t_f - t_i

Where,

‘Δv→’ is the change in velocity,

‘Δt’ is the time taken

Instantaneous Acceleration = limΔ t→0 Δv→/Δt = dv→/dt

• Acceleration is a vector quantity. (Velocity is a Vector)

• Acceleration can be either positive or negative.

Example Problem

The Position of an object moving along X - axis is given by −

x = 4 + 5t² - 2t³

Calculate the following −

• Velocity at t = 0s and t = 2s.
• Acceleration at t = 0s and t = 2s.
• Average Acceleration between t = 0s and t = 2s

Solution

x = 4 + 5t² - 2t³

Therefore,

• v = dx/dt = 10t - 6t²

• a = dv/dt = 10 - 12t

Velocity at t = 0s and t = 2s −

(v)_t=0 = (10t - 6t²)_t=0 = 0 m/s

(v)_t=2 = (10t - 6t²)_t=2 = -4 m/s

Acceleration at t = 0s and t = 2s −

(a)_t=0 = (10 - 12t)_t=0 = 10 m/s²

(a)_t=2 = (10 - 12t)_t=2 = -12 m/s²

Average Acceleration between t = 0s and t = 2s −

Average Acceleration = v_f - v_i/t_f - t_i = (v)_t=2 - (v)_t=0/2 - 0

Average Acceleration = (-12) - (10)/2 = -11 m/s²

Important Points

When velocity and acceleration have the same direction (or same sign), magnitude of velocity increases. Object speeds up.

When velocity and acceleration have opposite directions (or opposite signs), the magnitude of velocity decreases. Object slows down.

This is phenomenon of slowing down is called Retardation. Also, it should never be confused with negative acceleration.