What is the force acting on the particle if the motion of the particle is given as : y = ut + (1/2) gt²

The correct option is F = mg.

Step-by-step Explanation:

1. The position of the particle as a function of time is given by the equation of motion:
$$y=ut+\frac{1}{2}g{t}^2$$
where
$y$
is the displacement,
$u$
is the initial velocity,
$g$
is the acceleration due to gravity, and
$t$
is the time.

2. To find the velocity of the particle, we differentiate the displacement with respect to time:
$$v=\frac{dy}{dt}$$
Substituting the value of
$y$
into the differentiation:
$$v=\frac{d}{dt}\left(ut+\frac{1}{2}g{t}^2\right)=u+gt$$

3. Next, to find the acceleration of the particle, we differentiate the velocity with respect to time:
$$a=\frac{dv}{dt}$$
Substituting the expression for velocity:
$$a=\frac{d}{dt}\left(u+gt\right)=g$$

4. According to Newton's second law of motion, the force acting on a particle of mass
$m$
is the product of its mass and its acceleration:
$$F=ma$$

5. Substituting the calculated acceleration
$a=g$
into the formula for force, we get:
$$F=mg$$
Therefore, the force acting on the particle is
$F=mg$
.