Question
A particle moves along the -axis under the variable force , where is in metres. Find the work done by this force as the particle moves from to . Then, if the particle has mass and starts from rest at , find its speed at .
Solution — Step by Step
Force is variable — it depends on . So does not apply. We have to use the integral definition: . This is the trap of the “work” chapter — students see “force × distance” and forget the formula breaks for non-constant .
. So:
, .
Why This Works
The work-energy theorem holds for any force — constant, variable, or even non-conservative. As long as we compute net work correctly (here, only one force acts, so net work = work by ), the change in KE follows directly.
Integration replaces multiplication when the integrand varies. We are summing over infinitesimal slices of the path.
Alternative Method
If were a conservative force, we could find a potential energy such that , and compute . Here (up to a constant), giving . Same answer — useful when you have to track energy across multiple stages.
Common Mistake
Two trap variants. (1) Plugging the average force and multiplying by — this works only for linear , not quadratic. (2) Forgetting that the particle starts from rest; the theorem says , not . If , you must add the initial KE.