Find the position of the particle.

Question

A particle is moving with the given data.

a(t) = 2t + 3,    s(0) = 3,    v(0) = −2

s(t) = 

Answer 1

By definition,

v = ds/dt

a = dv/dt

Therefore we obtain v and s from a by integration

v(t) = ∫a(t)dt = ∫(2t+3)dt = t² + 3t + C

Using the constraint v(0) = -2

0² + 3×0 + C = -2

C = -2

Therefore

v(t) = t² + 3t - 2

s(t) = ∫v(t)dt = ∫(t² + 3t - 2)dt = t³/3 + 3t²/2 - 2t + D

Using the constraint s(0) = 3

0³/3 + 3×0²/2 - 2×0 + D = 3

D = 3

Therefore

s(t) = t³/3 + 3t²/2 - 2t + 3