The relation between time \( t \) and distance \( x \) is \( t = \alpha x^2 + \beta x \), where \( \alpha \) and \( \beta \) are constants. The relation between acceleration \( a \) and velocity \( v \) is:
- \( a = -2 \alpha v^3 \)
- \( a = -5 \alpha v^5 \)
- \( a = -3 \alpha v^2 \)
- \( a = -4 \alpha v^4 \)
The Correct Option is A
Approach Solution - 1
To find the relation between acceleration \( a \) and velocity \( v \), let's start from the given relation between time \( t \) and distance \( x \):
\(t = \alpha x^2 + \beta x\)
We need to determine the expressions for velocity and acceleration. Velocity \( v \) is given by:
\(v = \frac{dx}{dt}\)
First, express \( x \) as a function of \( t \) to find the derivative:
\(\frac{dt}{dx} = 2\alpha x + \beta\)
Thus, the velocity \( v \) can be written as:
\(v = \frac{dx}{dt} = \frac{1}{\frac{dt}{dx}} = \frac{1}{2\alpha x + \beta}\)
Now, let's find the acceleration \( a \), which is the derivative of velocity with respect to time:
\(a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \cdot \frac{dv}{dx}\)
Apply the chain rule by differentiating \( v \) with respect to \( x \):
\(v = \frac{1}{2\alpha x + \beta} \implies \frac{dv}{dx} = -\frac{2\alpha}{(2\alpha x + \beta)^2}\)
Therefore, substitute \( \frac{dv}{dx} \) and \( v \) to find \( a \):
\(a = v \cdot \frac{dv}{dx} = \left(\frac{1}{2\alpha x + \beta}\right) \cdot \left(-\frac{2\alpha}{(2\alpha x + \beta)^2}\right) = -\frac{2\alpha}{(2\alpha x + \beta)^3}\)
Substitute \( v = \frac{1}{2\alpha x + \beta} \) back into the expression for acceleration:
\(a = -2\alpha \left(\frac{1}{2\alpha x + \beta}\right)^3 = -2\alpha v^3\)
Thus, the correct relation between acceleration \( a \) and velocity \( v \) is:
\(a = -2 \alpha v^3\)
Therefore, the correct answer is \(a = -2 \alpha v^3\).
Approach Solution -2
Given:
\[ t = \alpha x^2 + \beta x \]
Differentiating with respect to \( x \):
\[ \frac{dt}{dx} = 2\alpha x + \beta \]
Using:
\[ \frac{1}{v} = 2\alpha x + \beta \quad \implies \quad v = \frac{1}{2\alpha x + \beta} \]
Differentiating with respect to time:
\[ -\frac{1}{v^2} \frac{dv}{dt} = 2\alpha \quad \implies \quad \frac{dv}{dt} = -2\alpha v^3 \]
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