The relation between time t and distance x is `t=ax^(2)+bx` where a and b are constants. The acceleration and velocity relation is acceleration `= - k av^(3)`. The value of 'k' is

Answer

To find the value of 'k' in the given problem, we will follow these steps:

### Step 1: Differentiate the relation between time and distance
We start with the equation given in the problem:
\[ t = ax^2 + bx \]
We differentiate this equation with respect to time \( t \):
\[ \frac{dt}{dt} = \frac{d}{dt}(ax^2 + bx) \]
Using the chain rule, we get:
\[ 1 = 2ax \frac{dx}{dt} + b \frac{dx}{dt} \]
Let \( v = \frac{dx}{dt} \) (the velocity), so we can rewrite this as:
\[ 1 = (2ax + b)v \]
This is our equation (1).

### Step 2: Differentiate with respect to distance
Next, we differentiate the equation \( 1 = (2ax + b)v \) with respect to \( x \):
\[ 0 = \frac{d}{dx}((2ax + b)v) \]
Applying the product rule:
\[ 0 = (2a)v + (2ax + b)\frac{dv}{dx} \]

### Step 3: Substitute acceleration relation
We know that acceleration \( a \) can be expressed as:
\[ a = v \frac{dv}{dx} \]
Substituting this into our equation gives:
\[ 0 = 2av + (2ax + b)\frac{dv}{dx} \]

### Step 4: Solve for \( \frac{dv}{dx} \)
Rearranging the equation:
\[ 0 = 2av + (2ax + b)\frac{dv}{dx} \]
This implies:
\[ (2ax + b)\frac{dv}{dx} = -2av \]
Thus,
\[ \frac{dv}{dx} = -\frac{2av}{2ax + b} \]

### Step 5: Substitute into acceleration relation
From the acceleration relation given in the problem:
\[ a = -k av^3 \]
Substituting \( a = v \frac{dv}{dx} \):
\[ v \frac{dv}{dx} = -k av^3 \]
Substituting \( \frac{dv}{dx} \) from the previous step:
\[ v \left(-\frac{2av}{2ax + b}\right) = -k av^3 \]

### Step 6: Simplify and solve for \( k \)
Cancelling \( v \) (assuming \( v \neq 0 \)):
\[ -\frac{2a}{2ax + b} = -k av^2 \]
Thus,
\[ \frac{2}{2ax + b} = k v^2 \]
From equation (1), we have:
\[ 2ax + b = \frac{1}{v} \]
Substituting this into our equation gives:
\[ \frac{2}{\frac{1}{v}} = k v^2 \]
This simplifies to:
\[ 2v = k v^2 \]
Dividing both sides by \( v \) (assuming \( v \neq 0 \)):
\[ k = \frac{2}{v} \]

### Conclusion
Thus, the value of \( k \) is:
\[ k = 2 \]