For motion of an object, the velocity “v” depends upon displacement “x” as, v = 3x² -4x . Then what is the acceleration, at x = 2.

To find the acceleration of an object when its velocity depends on displacement as given by the equation \( v = 3x^2 - 4x \), we need to follow a few steps. Acceleration is defined as the rate of change of velocity with respect to time. However, since we have velocity as a function of displacement, we can use the chain rule to express acceleration in terms of displacement.

Understanding the Relationship Between Velocity and Acceleration

Acceleration \( a \) can be expressed as:

• \( a = \frac{dv}{dt} \) (directly from the definition of acceleration)
• Using the chain rule, we can rewrite this as \( a = \frac{dv}{dx} \cdot \frac{dx}{dt} \), where \( \frac{dx}{dt} \) is the velocity \( v \).

Thus, we can express acceleration as:

\( a = v \cdot \frac{dv}{dx} \)

Step-by-Step Calculation

Now, let's calculate the acceleration at \( x = 2 \). First, we need to find \( \frac{dv}{dx} \).

1. Differentiate the Velocity Function

Given \( v = 3x^2 - 4x \), we differentiate it with respect to \( x \):

\( \frac{dv}{dx} = \frac{d}{dx}(3x^2 - 4x) = 6x - 4 \)

2. Evaluate \( \frac{dv}{dx} \) at \( x = 2 \)

Now, substituting \( x = 2 \) into the derivative:

\( \frac{dv}{dx} = 6(2) - 4 = 12 - 4 = 8 \)

3. Find Velocity at \( x = 2 \)

Next, we need to find the velocity at \( x = 2 \):

\( v = 3(2^2) - 4(2) = 3(4) - 8 = 12 - 8 = 4 \)

4. Calculate Acceleration

Now we can find the acceleration using the formula:

\( a = v \cdot \frac{dv}{dx} \)

Substituting the values we found:

\( a = 4 \cdot 8 = 32 \)

Final Result

Therefore, the acceleration of the object at \( x = 2 \) is \( 32 \, \text{m/s}^2 \). This means that at this point in its motion, the object is accelerating quite rapidly, which could indicate a significant change in its velocity over time.