Constant acceleration with algebraic unknowns

3. A car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The car is modelled as a particle.
At time \(t = 0\), the car is at the point \(A\) and the driver sees a road sign 48 m ahead.
Let \(t\) seconds be the time that elapses after the car passes \(A\).
In a first model, the car is assumed to decelerate uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) from \(A\) until the car reaches the road sign.

1. Use this first model to find the speed of the car as it reaches the sign. The road sign indicates that the speed limit immediately after the sign is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   In a second model, the car is assumed to decelerate uniformly at \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(A\) until it reaches a speed of \(13 \mathrm {~ms} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
2. Use this second model to find the value of \(t\) at which the car reaches the sign. In a third model, the car is assumed to move with constant speed \(25 \mathrm {~ms} ^ { - 1 }\) from \(A\) until time \(t = 0.2\), the car then decelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
3. Use this third model to find the value of \(t\) at which the car reaches the sign.

A car moves in a straight line with initial speed \(u\) m s\(^{-1}\) and constant acceleration \(a\) m s\(^{-2}\). The car takes 5 s to travel the first 80 m and it takes 8 s to travel the first 160 m. Find \(a\) and \(u\). [6]

A car travels along a straight road with constant acceleration. It passes through points \(A\), \(B\) and \(C\). The car passes point \(A\) with velocity 14 m s\(^{-1}\). The two sections \(AB\) and \(BC\) are of equal length. The times taken to travel along \(AB\) and \(BC\) are 5 s and 3 s respectively.

1. Write down an expression for the distance \(AB\) in terms of the acceleration of the car. Write down a similar expression for the distance \(AC\). Hence show that the acceleration of the car is 4 m s\(^{-2}\). [4]
2. Find the speed of the car as it passes point \(C\). [2]

A car is moving along a straight horizontal road with constant acceleration \(a\) m s\(^{-2}\) (\(a > 0\)). At time \(t = 0\) the car passes the point \(P\) moving with speed \(u\) m s\(^{-1}\). In the next 4 s, the car travels 76 m and then in the following 6 s it travels a further 219 m. Find

1. the value of \(u\),
2. the value of \(a\).

[7]

Three posts \(P\), \(Q\) and \(R\) are fixed in that order at the side of a straight horizontal road. The distance from \(P\) to \(Q\) is 45 m and the distance from \(Q\) to \(R\) is 120 m. A car is moving along the road with constant acceleration \(a\) m s\(^{-2}\). The speed of the car, as it passes \(P\), is \(u\) m s\(^{-1}\). The car passes \(Q\) two seconds after passing \(P\), and the car passes \(R\) four seconds after passing \(Q\). Find

1. the value of \(u\),
2. the value of \(a\).

[7]

A lorry is moving along a straight horizontal road with constant acceleration. The lorry passes a point \(A\) with speed \(u \text{ m s}^{-1}\), \((u < 34)\), and 10 seconds later passes a point \(B\) with speed \(34 \text{ m s}^{-1}\). Given that \(AB = 240\) m, find

1. the value of \(u\), [3]
2. the time taken for the lorry to move from \(A\) to the mid-point of \(AB\). [6]