A particle of mass m moves from rest a time t=0, under the action of a variable force f(t) = A*t*exp(-B*t), where A,B are positive constants. Find the speed of the particle for large t, expressing the answer in terms of m, A, and B.
First, we need to find a formula for the speed of the particle, v, at a given time, t. Relate the speed, v, to the force f, using Newton's 2nd Law: f(t) = mdv/dt (remembering that acceleration is the time-derivative of speed). The resulting equation can be solved using integration by parts [run through on whiteboard?]. Make sure to keep track of minus signs, and to include a constant of integration + c. This gives mv(t) = - A/(B^2) * (1 + Bt) * exp(-Bt) +c. To find the value of c, note that the particle starts from rest; so, we can use the condition v=0 at t=0. Substituting this in gives c = A/(B^2). Noting that the exponential term decays to zero for large t, the final speed is vfinal= A/(m*(B^2)). [Possible extra Qs: units of A,B ; sketch v]
