Which Solution is Incorrect and Why?

[misterpickle]

I'm a TA and there is an inconsistency in what the solutions manual states and what the students are turning in, but both seem correct.

The Latex code isn't working properly for the post, but it is easy to see what equations are used

Homework Statement

A particle of mass m moves along the x axis. Its position varies with time according to $$x=2t^3 - 4t^2$$ , where x is in meters and t is in seconds. Find

a. The velocity and acceleration of the particle as functions of t.

b. The power delivered to the particle as a function of t

c. the work done by the net force from $$t=0 to t=t_{1}$$

Homework Equations

The first 2 parts are easy and the answers are given by
$$v=6t^2 - 8t ; a=12t - 8 ; P=power=8mt(9t^2 - 18t + 8)$$

The Attempt at a Solution

The problem is that both solutions are correct mathematically, but they are different. Therefor one must be wrong and I need someone to tell me which on, and WHY?

Solution manual
$$w=\int_0^t \! P \, dt<br /> w=\int_0^t \! 8m(9t^3 - 18t^2 + 8t) \, dt<br /> w=2mt^{2}(9t^2-24t+16)$$
Student Solution
$$w=F\Delta x=ma\Delta x = m(12t - 8)(2t^3 - 4t^2)$$
This becomes,
$$w=m(24t^4 - 64t^3 + 32t)$$
Which one of these is wrong, since the math is right on both of them and the equations are valid for power?

 

[marmoset]

The second equation (Work = force*distance moved) is only valid for constant forces, in fact you can derive it from the first equation by letting P = F*dx/dt for some constant F.

 

[misterpickle]

So the student solution is incorrect?

 

[Redbelly98]

Student Solution
$$w=F\Delta x=ma\Delta x = m(12t - 8)(2t^3 - 4t^2)$$

So the student solution is incorrect?

Yes, it is incorrect. The correct equation is

W = ∫ F dx​

Which, as marmoset said, equals F Δx only for a constant force.

 

[Liquidxlax]

So the student solution is incorrect?

students looks wrong to me