Question

The acceleration of a particle increases according to the equation a=bt over time. If the initial velocity of the particle is vo, then will be the displacement of the particle in time be t?

Answer 1

To find the displacement of the particle over time, we need to integrate the equation for acceleration with respect to time, and then solve for displacement.

Given: Acceleration (a) = bt Initial velocity (vo)

Let's integrate the equation for acceleration with respect to time (t) to obtain the equation for velocity (v):

∫a dt = ∫(bt) dt v = ∫(bt) dt v = (1/2)bt^2 + C

Here, C represents the constant of integration. To determine this constant, we'll use the initial velocity (vo) at time t = 0:

v(0) = (1/2)b(0)^2 + C vo = C

So, C = vo, and the equation for velocity becomes:

v = (1/2)bt^2 + vo

Now, we integrate the equation for velocity with respect to time (t) to obtain the equation for displacement (s):

∫v dt = ∫[(1/2)bt^2 + vo] dt s = (1/6)bt^3 + vot + D

Here, D represents the constant of integration. To determine this constant, we'll use the initial displacement (so) at time t = 0:

s(0) = (1/6)b(0)^3 + vo(0) + D so = D

So, D = so, and the equation for displacement becomes:

s = (1/6)bt^3 + vot + so

Therefore, the displacement of the particle at time t will be given by the equation:

s = (1/6)bt^3 + vot + so