a particle moves along a line so that it’s acceleration for t > 0 is given by a(t) = t+3/squa. t^3 +1. If the particles velocity at t=0 is 5 what is the velocity of the particle at t=3.

Answer 1

To find the velocity of the particle at t=3, we first need to find the expression for the particle's velocity as a function of time by integrating the acceleration function:

a(t) = t + 3/sqrt(t^3 + 1)

To find the velocity function v(t), we integrate the acceleration function with respect to t:

v(t) = ∫(t + 3/sqrt(t^3 + 1)) dt
v(t) = 1/2 * t^2 + 6 * sqrt(t^3 + 1) + C

Given that the velocity at t=0 is 5, we can solve for the constant C:

v(0) = 1/2 * 0^2 + 6 * sqrt(0^3 + 1) + C
5 = 6 + C
C = -1

So, the velocity function is:

v(t) = 1/2 * t^2 + 6 * sqrt(t^3 + 1) - 1

To find the velocity of the particle at t=3, we substitute t=3 into the velocity function:

v(3) = 1/2 * 3^2 + 6 * sqrt(3^3 + 1) - 1
v(3) = 9/2 + 6 * sqrt(28) - 1
v(3) ≈ 9/2 + 16.73 - 1
v(3) ≈ 24.23

Therefore, the velocity of the particle at t=3 is approximately 24.23.