The velocity of a body depends on time according to the equation `v=(t^(2))/(10)+20`. The body is undergoing

Options

A. Uniform acceleration
B. Uniform retardation
C. Non - uniform acceleration
D. Zero acceleration

Answer

To determine the nature of motion of the body described by the velocity equation \( v = \frac{t^2}{10} + 20 \), we need to analyze the acceleration of the body. Here are the steps to solve the problem:

### Step 1: Identify the given equation for velocity
The velocity of the body is given by:
\[ v = \frac{t^2}{10} + 20 \]

### Step 2: Differentiate the velocity equation to find acceleration
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. Therefore, we differentiate the velocity equation \( v \) with respect to \( t \):
\[ a = \frac{dv}{dt} \]

### Step 3: Differentiate the velocity function
Using the given velocity equation, we differentiate:
\[ \frac{dv}{dt} = \frac{d}{dt}\left(\frac{t^2}{10} + 20\right) \]
The derivative of \( \frac{t^2}{10} \) is \( \frac{2t}{10} = \frac{t}{5} \), and the derivative of the constant \( 20 \) is \( 0 \). Thus:
\[ a = \frac{t}{5} \]

### Step 4: Analyze the acceleration
From the result \( a = \frac{t}{5} \), we can see that acceleration is directly proportional to time \( t \). This means that as time increases, acceleration also increases.

### Step 5: Determine the nature of motion
1. **Uniform acceleration**: This would mean that acceleration is constant. Since \( a = \frac{t}{5} \) is not constant (it increases with time), this is not the case.
2. **Retardation**: This occurs when acceleration is negative or decreasing. Since \( a \) is increasing, this is also not the case.
3. **Non-uniform acceleration**: Since acceleration is changing with time, this is indeed the case.

### Conclusion
The body is undergoing **non-uniform acceleration**.

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