The motion of a particle of mass m is described by `y =ut + (1)/(2) g t^(2)` . Find the force acting on the particale .

Options

A. 1.mg
B. 2.`(mu)/(t)`
C. 3.2mg
D. 4.`(2mu)/(t)`

Explanation

To solve the problem, we need to find the force acting on a particle of mass \( m \) whose motion is described by the equation:

\[ y = ut + \frac{1}{2} g t^2 \]

### Step-by-Step Solution:

1. **Identify the Equation of Motion**:
The given equation \( y = ut + \frac{1}{2} g t^2 \) describes the vertical motion of a particle under the influence of gravity, where:
- \( y \) is the vertical displacement,
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity,
- \( t \) is the time.

2. **Differentiate to Find Velocity**:
To find the velocity \( v \), we differentiate the displacement \( y \) with respect to time \( t \):
\[ v = \frac{dy}{dt} = \frac{d}{dt}(ut + \frac{1}{2} g t^2) \]
Using the power rule of differentiation:
\[ v = u + gt \]

3. **Differentiate to Find Acceleration**:
Now, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(u + gt) \]
Since \( u \) is a constant, its derivative is 0:
\[ a = 0 + g = g \]

4. **Apply Newton's Second Law**:
According to Newton's second law, the force \( F \) acting on an object is given by:
\[ F = ma \]
Substituting the value of acceleration \( a \):
\[ F = mg \]

5. **Conclusion**:
The force acting on the particle is:
\[ F = mg \]

### Final Answer:
The force acting on the particle is \( F = mg \).

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