A particle is moving with acceleration a(t)=24t+12. its position at time t=0 is s(0)=15 and its velocity at time t=0 is v(0)=17. What is its position at time t=14?

Answer 1

dv/dt = acceleration so:

dv/dt = 24t + 12

dv = (24t + 12) dt

∫dv = ∫(24t + 12) dt

v(t) = 12t² + 12t + C₁

Since v(0) = 17 we can say:

17 = 12(0)² + 12(0) + C₁ or C₁ = 17

So v(t) = 12t² + 12t + 17

ds/dt = v = 12t² + 12t + 17 so:

ds = (12t² + 12t + 17) dt

∫ds = ∫(12t² + 12t + 17) dt

s(t) = 4t³ + 6t² + 17t + C₂

Since s(0) = 15 we can say:

15 = 4(0)³ + 6(0)² + 17(0) + C₂ or C₂ = 15

So s(t) = 4t³ + 6t² + 17t + 15

Finally, s(14) = 4(14)³ + 6(14)² + 17(14) + 15 = 12,405 (units unknown)