A force F_1 acts on a particle, accelerating it from rest to a...

Let the mass of the particle be $m$.

During the acceleration phase, the particle starts from rest and attains a velocity $v$. Let $t_1$ be the time for which the force $F_1$ acts. Using the impulse-momentum theorem, we have:

$F_1 t_1 = m v - 0 = m v$

During the deceleration phase, the particle is brought back to rest from velocity $v$. Let $t_2$ be the time for which the force $F_2$ acts. The magnitude of the change in momentum is the same. Thus, we have:

$F_2 t_2 = m v - 0 = m v$

Equating the two expressions, we get:

$F_1 t_1 = F_2 t_2$

Since the time intervals $t_1$ and $t_2$ are not specified in the problem, they can take any positive values.

If $t_1 = t_2$, then $F_1 = F_2$.

If $t_1 \neq t_2$, then $F_1 \neq F_2$.

Therefore, the magnitude of the force $F_1$ may or may not be equal to $F_2$. The correct statement is that $F_1$ may be equal to $F_2$.