The distance travelled by a body is proportional to the square of time. What type of motion this body has ?

Options

A. projectile motion
B. uniform accelerated motion
C. Non uniform accelerated motion
D. none

Explanation

To determine the type of motion of a body whose distance travelled is proportional to the square of time, we can follow these steps:

### Step 1: Establish the relationship between distance and time
Given that the distance \( S \) travelled by the body is proportional to the square of time \( t \), we can express this relationship mathematically as:
\[ S \propto t^2 \]
This can be rewritten with a constant \( C \):
\[ S = Ct^2 \]

### Step 2: Differentiate to find velocity
To find the velocity \( v \) of the body, we differentiate the distance \( S \) with respect to time \( t \):
\[ v = \frac{dS}{dt} = \frac{d}{dt}(Ct^2) \]
Using the power rule of differentiation:
\[ v = 2Ct \]

### Step 3: Differentiate to find acceleration
Next, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(2Ct) \]
Again, applying the power rule:
\[ a = 2C \]

### Step 4: Analyze the results
From the calculations:
- The velocity \( v = 2Ct \) shows that the velocity is linearly dependent on time.
- The acceleration \( a = 2C \) is a constant value.

### Conclusion
Since the acceleration is constant, we conclude that the body is undergoing uniformly accelerated motion. Thus, the type of motion exhibited by the body is **uniformly accelerated motion**.

---