Particle Moves Along The X Axis

Question

I asked this question before:

A 5.60-kg particle moves along the x axis. Its position varies with time according to x = t + 4.0t³, where x is in meters and t is in seconds.

(a) Find the kinetic energy of the particle at any time t. (Use the following as necessary: t.)

(b) Find the magnitude of the acceleration of the particle and the force acting on it at time t. (Use the following as necessary: t.)

(c) Find the power being delivered to the particle at time t. (Use the following as necessary: t.)

(d) Find the work done on the particle in the interval t = 0 to t = 1.50 s.

I can work out part b and c but I have no idea how to determine the part a) equation. The answer I came up with was: kinetic energy = 403.2*t^4

this is wrong, the answer should have kinetic energy = 2.8+__+403.2*t^4 but I have no idea how to get to that point, would anyone please be able to explain the process to getting to that answer please?

Answer 1

Since x = t + 4t³ then v = dx/dt = 1 + 12t² KE = mv²/2 so KE = 5.6 ( 1 + 12t²)²/2

hence KE = 2.8(1 +24t + 144t⁴) so KE = 2.8 + 67.2t + 403.2t⁴