12 A particle moves on a straight line with a constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The initial velocity of the particle is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
After \(T\) seconds the particle has velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
This information is shown on the velocity-time graph.
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-18_602_1065_813_541}
The displacement, \(S\) metres, of the particle from its initial position at time \(T\) seconds is given by the formula
$$S = \frac { 1 } { 2 } ( U + V ) T$$
12
- By considering the gradient of the graph, or otherwise, write down a formula for \(a\) in terms of \(U , V\) and \(T\).
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[1 mark]
12 - Hence show that \(V ^ { 2 } = U ^ { 2 } + 2 a S\)
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[3 marks]