Displacement of a particle moving in a straight line is represented as follows:
`x=at^(3)+bt^(2)+ct+d` Ratio of initial velocity to initial acceleration depends.

Options

A. Only on a and b
B. Only on b and c
C. Only on c and d
D. Only on d and a

Explanation

To find the ratio of initial velocity to initial acceleration for the given displacement function \( x = at^3 + bt^2 + ct + d \), we will follow these steps:

### Step 1: Find the expression for velocity
The velocity \( v \) is the first derivative of displacement \( x \) with respect to time \( t \).

\[ v = \frac{dx}{dt} = \frac{d}{dt}(at^3 + bt^2 + ct + d) \]

### Step 2: Differentiate the displacement function
Using the power rule of differentiation, we differentiate each term:

- The derivative of \( at^3 \) is \( 3at^2 \)
- The derivative of \( bt^2 \) is \( 2bt \)
- The derivative of \( ct \) is \( c \)
- The derivative of the constant \( d \) is \( 0 \)

Thus, the expression for velocity becomes:

\[ v = 3at^2 + 2bt + c \]

### Step 3: Calculate initial velocity
The initial velocity \( v_0 \) is the value of velocity at \( t = 0 \):

\[ v_0 = v(0) = 3a(0)^2 + 2b(0) + c = c \]

### Step 4: Find the expression for acceleration
Acceleration \( a \) is the derivative of velocity \( v \) with respect to time \( t \):

\[ a = \frac{dv}{dt} = \frac{d}{dt}(3at^2 + 2bt + c) \]

### Step 5: Differentiate the velocity function
Using the power rule again, we differentiate each term:

- The derivative of \( 3at^2 \) is \( 6at \)
- The derivative of \( 2bt \) is \( 2b \)
- The derivative of \( c \) is \( 0 \)

Thus, the expression for acceleration becomes:

\[ a = 6at + 2b \]

### Step 6: Calculate initial acceleration
The initial acceleration \( a_0 \) is the value of acceleration at \( t = 0 \):

\[ a_0 = a(0) = 6a(0) + 2b = 2b \]

### Step 7: Find the ratio of initial velocity to initial acceleration
Now, we can find the ratio of initial velocity \( v_0 \) to initial acceleration \( a_0 \):

\[ \frac{v_0}{a_0} = \frac{c}{2b} \]

### Conclusion
The ratio of initial velocity to initial acceleration depends on the constants \( c \) and \( b \) in the displacement equation.

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