If displacement `x= t^(4)` , then the ratio of acceleration and velocity of motion will be

Options

A. `(3)/(t)`
B. `(t)/(3)`
C. 3t
D. `(4)/(t)`

Explanation

To solve the problem, we need to find the ratio of acceleration to velocity given the displacement equation \( x = t^4 \).

### Step 1: Find the velocity
The velocity \( v \) is defined as the rate of change of displacement with respect to time. We can find the velocity by differentiating the displacement \( x \) with respect to time \( t \).

\[ v = \frac{dx}{dt} \]

Given \( x = t^4 \), we differentiate:

\[ v = \frac{d(t^4)}{dt} = 4t^3 \]

### Step 2: Find the acceleration
The acceleration \( a \) is defined as the rate of change of velocity with respect to time. We can find the acceleration by differentiating the velocity \( v \) with respect to time \( t \).

\[ a = \frac{dv}{dt} \]

Now, we differentiate \( v = 4t^3 \):

\[ a = \frac{d(4t^3)}{dt} = 12t^2 \]

### Step 3: Find the ratio of acceleration to velocity
Now that we have both the acceleration and velocity, we can find the ratio \( \frac{a}{v} \):

\[ \frac{a}{v} = \frac{12t^2}{4t^3} \]

### Step 4: Simplify the ratio
We can simplify the ratio:

\[ \frac{a}{v} = \frac{12}{4} \cdot \frac{t^2}{t^3} = 3 \cdot \frac{1}{t} = \frac{3}{t} \]

### Final Answer
The ratio of acceleration to velocity is:

\[ \frac{a}{v} = \frac{3}{t} \]