Chapter: Kinematics of particle and rigid bodies· Engineering Mechanics

The acceleration of a particle in rectilinear motion is given by \(a=64-4x^2\ \text{m/s}^2\), where x is in metres. At \(x=0\) the particle is at rest. Find the velocity at \(x=3\) m, and the position and value of the maximum velocity.

Q.12 The acceleration of a particle in rectilinear motion is given by \(a=64-4x^2\ \text{m/s}^2\), where x is in metres. At \(x=0\) the particle is at rest. Find the velocity at \(x=3\) m, and the position and value of the maximum velocity.

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Answer: \(v=17.66\) m/s at \(x=3\) m; maximum velocity \(=18.48\) m/s at \(x=4\) m.

Since a is given as a function of x, use \(a=v\dfrac{dv}{dx}\):

\[\int_0^v v\,dv=\int_0^x(64-4x^2)\,dx\Rightarrow\frac{v^2}{2}=64x-\frac{4x^3}{3}\Rightarrow v^2=128x-\frac{8x^3}{3}\]

At \(x=3\) m: \(v^2=384-72=312\Rightarrow v=17.66\) m/s.

Maximum velocity: v is maximum when \(\dfrac{dv}{dx}=0\), i.e. \(a=0\): \(64-4x^2=0\Rightarrow x=4\) m.

\(v_{max}^2=128(4)-\dfrac{8(64)}{3}=512-170.67=341.33\Rightarrow v_{max}=18.48\) m/s.

Doubtly AI · 27 Oct 2026 AI-verified · Oct 2026 Report

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The acceleration of a particle in rectilinear motion is given by \(a=64-4x^2\ \text{m/s}^2\), where x is in metres. At \(x=0\) the particle is at rest. Find the velocity at \(x=3\) m, and the position and value of the maximum velocity.

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