The acceleration of a particle is expressed as a = 10 - x. The particle starts (time = 0) with no initial velocity at x = 0. What will be the velocity of the particle when acceleration is zero?

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BHEL Engineer Trainee Mechanical 24 Aug 2023 Official Paper

Answer 1

Concept:

We analyze the motion of a particle with variable acceleration to determine its velocity when the acceleration becomes zero.

Given:

Step 1: Find the position when acceleration is zero

Set \( a = 0\)

\( 10 - x = 0 \)

\( x = 10 \, \text{m} \)

Step 2: Relate acceleration to velocity and position

Using the chain rule, acceleration can be expressed as:

\( a = \frac{dv}{dt} = v \frac{dv}{dx} = 10 - x \)

Separate variables and integrate:

\( v \, dv = (10 - x) \, dx \)

Step 3: Integrate both sides

\( \int_{0}^{v} v \, dv = \int_{0}^{10} (10 - x) \, dx \)

\( \frac{v^2}{2} \bigg|_{0}^{v} = 10x - \frac{x^2}{2} \bigg|_{0}^{10} \)

\( \frac{v^2}{2} = 100 - 50 = 50 \)

\( v^2 = 100 \)

\( v = 10 \, \text{m/s} \)