Explanation

A particle moves on a line with a given velocityv(t) = p + q t + r t². Differentiate for acceleration and integrate for displacement from rest at the origin.r = 0 is constant acceleration; r = 0 and q = 0 is constant velocity.

A quadratic velocity has a simpler derivative and a cubic integral. The constant of integration is fixed by s = 0 at t = 0.

• a(t) = dv/dt = q + 2r t
• s(t) = ∫v dt = p t + ½q t² + ⅓r t³

If v later becomes zero the particle is instantaneously at rest and then reverses, provided a is not also zero.

Worked example

A particle moves on a straight line with v(t) = 3 + 2t − t², where v is in m/s and t is in seconds. The displacements is zero when t = 0. Find expressions fora(t) and s(t), and find v, aand s when t = 1 s.

1. a(t) = 2 − 2t
2. s(t) = 3t + t² − t³/3
3. When t = 1: v = 4 m/s, a = 0, s = 3 + 1 − ⅓ = 3.67 m

Show that the particle is instantaneously at rest when t = 3 s, and find its displacement then.

1. v(3) = 3 + 6 − 9 = 0, so it is instantaneously at rest
2. a(3) = 2 − 6 = −4 m/s², so it then reverses
3. s(3) = 9 + 9 − 9 = 9 m

If the t² term were missing, the acceleration would be the constant 2 m/s² and s = 3t + t².