Compute the displacement of the particle:

The acceleration time diagram for particle is as specified below .Computed velocity of the particle at several time intervals, plot velocity time diagram and computed the displacement of the particle at several intervals. Suppose initial velocity v₀ = 0 and initial displacement s₀ = 5.30 cm.

Solution

We have

dv/ dt  = a,   or          dv = a dt

∴ v₁  - v₀  = Area of (a - t ) diagram between t = 0 and t =1 seconds

∴ v₁  = v₀  + 10 × 1                 v₀  = 0

= 10 cm / sec

Likewise,

∴ v₂  - v₀  = Area of (a - t ) diagram between t = 0 and t = 2 seconds

= v₀  + 10 × 2

= 20 cm / sec

And         v₃  = 30 cm / sec

We have ds/ dt  = v, or ds = v dt

or s₁  - s₀  = Area of (v - t ) diagram among t = 0 and t =1 seconds

∴ s1 = s0 + (½) × 1 × 10

= 5.30 + 5.0

= 10.30 cm .

s₂  - s₀  = Area of (v - t ) diagram among t = 0 and t = 2 seconds

         = 5.3 + (½) × 20 × 2

= 25.3 cm .

s₃  - s₀  = Area of (v - t ) diagram among t = 0 and t = 3 seconds

s₃  = 5.3 + (½) × 30 × 3 = 50.3 cm .