OCR MEI M1 (Mechanics 1)

1 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$ where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\).
2. Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
3. Find an expression in terms of \(t\) for the acceleration of the particle.
4. Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
5. Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
7. Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).

3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training.
Marie runs along a straight line at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 ,
a = 0 & \text { for } t > 4 . \end{array}$$

1. Show that Nina's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 ,
   v = 8 & \text { for } t > 4 . \end{array}$$
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
3. Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).

4 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start

• P is travelling directly towards Q with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• Q is instantaneously at rest and has an acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directly towards P .

P continues with the same velocity and Q continues with the same acceleration. The cars collide \(T\) seconds after the start.

1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start. At the start, P is 90 m from Q .
2. Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).

5 The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a particle moving along a straight line is given by $$v = 3 t ^ { 2 } - 12 t + 14$$ where \(t\) is the time in seconds.

1. Find an expression for the acceleration of the particle at time \(t\).
2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\).
3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times.
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