Question

A particle of mass m is at rest at the origin at time `t=0` It is subjected to a force `F(t)=F_(0)e^(-bt)` in the X-direction. Its speed `V(t)` is depicted by which of the following curves

Answer 1

As the force is exponentially decreasing its accelertion, rate of increase of velocity will decrease with time. Thus the graph of velocity will be an increasing curve with decreasing slope with time
`a=(F)/(m) = (F_(0))/(m)e^(-bt)rArr(dv)/(dt) (F_(0))/(m)e^(-bt) rArrunderset(0)overset(v)(int)dv=underset(0)overset(t)int(F_(0))/(m)e^(-bt)dt`
`rArrv=[[F_(0))/(m)((1)/(-b))e^(-bt)]_(0)^(t)=[[F_(0))/(m)((1)/(b))e^(-bt)]_(t)^(0)`
`=(F_(0))/(mb)(e^(0)-e^(-bt))=(F_(0))/(mb) (1-e^(-bt))`
So, velocity increases continuously and attains a maximum value `v_(max) = (F_(0))/(mb)` .