SUVAT simultaneous equations: find u and a

6 A car is driven with constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), along a straight road. Its speed when it passes a road sign is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down two equations connecting \(a\) and \(u\). Hence find the values of \(a\) and \(u\).
2. What distance does the car travel in the 5 seconds after passing the road sign? Section B (36 marks)

4 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h] \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).

1. Find the initial velocity and the acceleration of the particle.
2. Prove that the particle is never at P .

2 \begin{figure}[h] \end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, \(t\), is measured in seconds. The particle has constant acceleration, \(a \mathrm {~ms} ^ { - 2 }\). Initially it is at the point O and has velocity \(u \mathrm {~ms} ^ { - 1 }\).
When \(t = 2\), the particle is at A where OA is 12 m . The particle is also at A when \(t = 6\).

1. Write down two equations in \(u\) and \(a\) and solve them.
2. The particle changes direction when it is at B . Find the distance AB .

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the top of a smooth inclined plane of length \(2 d\) metres. After its projection \(P\) moves downwards along a line of greatest slope with acceleration \(4 \mathrm {~ms} ^ { - 2 }\). At the instant 3 s after projection \(P\) has moved half way down the plane. \(P\) reaches the foot of the plane 5 s after the instant of projection.

1. Form two simultaneous equations in \(u\) and \(d\), and hence calculate the speed of projection of \(P\) and the length of the plane.
2. Find the inclination of the plane to the horizontal.
3. Given that the contact force exerted on \(P\) by the plane has magnitude 6 N , calculate the mass of \(P\).

2 A particle travels with constant acceleration along a straight line. A and B are points on this line 8 m apart. The motion of the particle is as follows.

• Initially it is at A.
• After 32 s it is at B .
• When it is at B its speed is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is moving away from A .

In either order, calculate the acceleration and the initial velocity of the particle, making the directions clear.

11 \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).

1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
   (b) Find the acceleration of \(P\). \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
   It is given that
   \section*{OCR} \section*{Oxford Cambridge and RSA}

2 A lift is travelling upwards and accelerating uniformly. During a 5 second period, it travels 16 metres and the speed of the lift increases from \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. \(\quad\) Find \(u\).
2. Find the acceleration of the lift.

2. A particle passes through a point \(O\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) and moves in a straight line with constant acceleration \(3.6 \mathrm {~ms} ^ { - 2 }\) for \(t\) seconds until it reaches the point \(P\). The acceleration is then reduced to \(2 \mathrm {~ms} ^ { - 2 }\) and this is maintained for another \(t\) seconds until the particle passes the point \(Q\) with speed \(16 \mathrm {~ms} ^ { - 1 }\). Calculate

1. the time taken by the particle to travel from \(O\) to \(Q\),
2. the distance \(O Q\).

3 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h] \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).

1. Find the initial velocity and the acceleration of the particle.
2. Prove that the particle is never at P .

4 A car is driven with constant acceleration, \(a \mathrm {~m} \mathrm {~s} { } ^ { 2 }\), along a straight road. Its speed when it passes a road sign is \(u \mathrm {~ms} { } ^ { 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~ms} { } ^ { 1 }\).

1. Write down two equations connecting \(a\) and \(u\). Hence find the values of \(a\) and \(u\).
2. What distance does the car travel in the 5 seconds after passing the road sign?