The acceleration function (in m/s^2) and the initial velocity v(0) are given for a particle moving along a line

Question

The acceleration function (in m/s²) and the initial velocity v(0) are given for a particle moving along a line

a(t) = 2t + 4,    v(0) = −5,    0 ≤ t ≤ 3

(a) Find the velocity at time t.

v(t) =______ m/s

(b) Find the distance traveled during the given time interval.

=_______m

Answer 1

Simon

a(t) = dv/dt

dv/dt = 2t + 4

dv = (2t+4) dt

Integrate both sides

v = t² + 4t + C

v(0) = -5, so C = -5

so v = t² + 4t - 5

v = ds / dt

So (t² + 4t - 5) dt = ds

Integrate both sides

t³/3 + 2t² - 5t + C = s

Distance travelled = s(3) - s(0) = 27/3 + 2.9 - 5.3 + C - 0 - 0 + 0 - C = 12

Mike