Specialist2025 Exam 1 · Q3Tech-free5 marks

Particle with velocity t over root(t^2 + k)

A particle starts from rest at a fixed point $O$ and travels in a straight line.

The velocity, $v$ m s $^{-1}$ , of the particle at time $t$ seconds has equation $v (t) = \frac{t}{\sqrt{^{}}}$ , where is a positive constant and .

Use integration to show that the displacement, $x$ metres, of the particle relative to $O$ is given by $x(t) = \sqrt{t^2 + k} - \sqrt{k}$ .

1 mark

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Find the initial acceleration, in terms of $k$ , of the particle in m s $^{-2}$ .

2 marks

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Another particle starts at $O$ at the same time as the first particle and follows the same path.

Its position relative to $O$ is described by the equation $s(t) = t$ .

Three seconds after leaving $O$ the second particle is 1 m ahead of the first particle.

Find the value of $k$ .

2 marks

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