How much force is needed to accelerate and decelerate the same particle?

Question

Suppose a force F1 was required to accelerate a particle from rest to a velocity v.The force F1 is then replaced by F2 which decelerates it to rest.Now my question is whether F1 is equal to F2 or not?

I think that F1 must not be equal to F2.From F=ma the acceleration gained by the particle must be made equal to zero in order to bring it to rest.

Hence it will require an acceleration in the opposite direction of the same magnitude i.e deceleration.Hence the negative sign of a must be used for F2.

Since force is a vector both magnitude and direction must be same for F1 and F2 to be equal.But I don't know if my reasoning or my answer is correct.

Answer 1

Let's assume we have a 1-dimensional problem along the $x$ axis. Let's also assume we have a pair of constant forces which act sequentially:

• $\vec{F}_1=F_1\hat{i}$, acting from $t=0$ to $t=t_1$
• $\vec{F}_2=F_2\hat{i}$, acting from $t>t_1$ to $t=t_2$
• the initial and final velocities are zero

While $\vec{F}_1$ is acting, the object (of mass $m$) accelerates constantly, gaining positive $x$ velocity until at the end of the time interval it has velocity $v_1$: $$v_1=0+a_1 t_1.$$

Starting at this time, $\vec{F}_2$ acts and the object accelerates to zero velocity: $$0 = a_1 t_1 + a_2(t_2-t_1).$$

If we multiply both sides by the object mass, $m$, and the unit vector $\hat{i}$ and rearrange, we get $$\vec{F}_1t_1 = -\vec{F}_2(t_2-t_1).$$ From this we see

• the forces must be in opposite directions
• the forces do not have to have equal magnitudes, but the impulses ($\vec{F}\Delta t$), due to each force, must add to zero.

As a note for the future, the impulse is changing the momentum, and is part of the conservation of momentum law. Forces can be thought of as agents of change: they transfer momentum during time intervals and they transfer energy over distances (work).