A body is initially at rest.A net force F acting on it varies with time.Then prove that the body can not remain at rest.
Theoretical Proof : For a body to be in equilibrium or in a rest state the total force acting must be zero in case, the force isn't constant , that is, it varies over time then the net velocity is not constant either ( Note: an object has a constant velocity due to 0 net force, in that case constant velocity is zero) since velocity differs that means it has an acceleration which implies the body changed its direction or it has moved a distance and is not at rest anymore.
Mathematical Proof:---
Initial velocity = 0
now Force acting was constant or( zero net force)hence acceleration was constant and both were zero hence F= a= 0
now as F varies over time, then differentiation both sides of F=ma w.r.t time we get
df/dt= m da/dt
but, da/dt = v, so we get
df/dt = m*v now since m is constant and Force is not zero therefore velocity is a non zero value which shows theat the body is not at rest. Hope I helped you mate!
Mathematical Proof:---
Initial velocity = 0
now Force acting was constant or( zero net force)hence acceleration was constant and both were zero hence F= a= 0
now as F varies over time, then differentiation both sides of F=ma w.r.t time we get
df/dt= m da/dt
but, da/dt = v, so we get
df/dt = m*v now since m is constant and Force is not zero therefore velocity is a non zero value which shows theat the body is not at rest. Hope I helped you mate!