Did I Solve This Work with Variable Force Problem Correctly?

[suffian]

I just did the following textbook problem (from my Calculus book under the chapter review from Applications of Integrals) but came up with an answer that differs from the one at back of the book:

A particle of mass m starts from rest at time t = 0 and is moved along the x-axis with constant acceleration a from x=0 to x=h against a variable force of magnitude F(t)=t². Find the work done.

Calculus and Analytic Geometry by George. B Thomas

I reasoned as follows:
The force I need to apply at any given time t to maintain constant acceleration would need to be t² to cancel out the force acting against me plus ma to maintain a constant forward acceleration. In other words F(t) = t²+ma.

If the acceleration is always a and the particle starts from rest at x=0 then
x''(t) = a
x'(t) = at
x(t) = at²/2

This means that once the particle reaches a position x, the following amount of time has elapsed:
t(x) = sqrt(2x/a), t > 0

Which in turn leads to the force that needs to applied at a given position x:
F( sqrt(2x/a) ) = 2x/a + ma

We now use the work formula W= ∫_a..b F(x)dx :
W = ∫_0..h (2x/a + ma)dx
W = mah + h²/a

On the otherhand, the book gives the following result:
W = 4h/3 sqrt(3mh)

I don't see how they arrived at their results since it doesn't take into account the acceleration of the particle. Surely, the faster the particle is accelerated the work should take toward a limiting value of mah(force * distance) because the particle will reach its destination faster and therefore have less resistance to deal with (remember the resistance was t²). But the book doesn't ever consider the acceleration.

Has the book made an error, or have i misinterpreted the problem?

 

[lethe]

man, the units don t even work out right for the answer the book gave. that s a sure fire way to tell a wrong answer.

plus, i didn t see any mistakes in your work.

 

[tacman]

It is possible that there is an error in the book, or that they used a different approach to solve the problem. However, it is also possible that you may have made a mistake in your reasoning or calculations. It would be best to double check your work and compare it to the book's solution to see where the discrepancy lies. Sometimes, different approaches can lead to different solutions, so it is important to understand the problem and the given information thoroughly before attempting to solve it. It is always a good idea to seek help from a teacher or tutor if you are unsure about your solution or if it differs from the given answer.