1. The problem statement, all variables and given/known data A body of mass m accelerates uniformly from rest to a speed [tex]v_{f}[/tex] in time [tex]t_{f}[/tex] Show that the work done on the body as a function of time, in terms of [tex]v_{f}[/tex],[tex]t_{f}[/tex] is: [tex]\frac{1}{2}m\frac{v_{f}^2}{t_{f}^2} t^2[/tex] 2. Relevant equations 1)[tex] W=\int F*dx[/tex] 2)[tex]V_{f}=at[/tex] 3)[tex]F=m\frac{V}{t}[/tex] 3. The attempt at a solution Well I know I would start out integrating equation 1 with equation 3. Then what?
Since the acceleration is uniform (i.e. constant) So is the force and thus the integral is not required. You're on the right lines with equation 3, but you need to consider the distance travelled by the object aswell. Think of the uniform acceleration equations.
Ah of course I get it now: [tex]r=\frac{1}{2}at^2[/tex] [tex]W=F*r \longrightarrow \frac{1}{2}ma^2t^2 \longrightarrow \frac{1}{2}m\frac{v^2}{t_{f}^2}t^2[/tex]
So the thing to remember from this about work is that if the force is constant then its simply the force multiplied by the distance (or in vector terms the dot product) and if the force varies you have to integrate.