Kinematics: displacement, velocity and acceleration Problem set A

A particle moves in a straight line. The displacement from a fixed point is \(s\) m, where \(s=2 t^{3} - 9 t^{2} + 12 t + 1\) and \(t\ge0\) is in seconds.

1. (a) Find the velocity when \(t=3\).
2. (b) Find the acceleration when \(t=3\).

A trolley moves along a straight track. The displacement from a fixed point is \(s=15e^{-0.3t}+t\) m, where \(t\ge0\) is in seconds. Give your answers to 3 significant figures.

1. (a) Find the velocity when \(t=2\).
2. (b) Find the acceleration when \(t=2\).

A particle moves in a straight line. The displacement from a fixed point is \(s=2 t^{3} - 15 t^{2} + 24 t\) m, for \(t\ge0\) seconds.

1. (a) Find the times when the particle is at rest.
2. (b) Find the displacement at each of those times.

A mass on a spring moves up and down. The displacement from its rest position is \(x=4\sin(3t)\) m, where \(t\) is in seconds. Use radians.

1. (a) Find the velocity when \(t=1\). Give your answer to 3 significant figures.
2. (b) Find the first time after \(t=0\) when the object is at rest.
3. (c) Write down the greatest speed of the object.

A particle moves in a straight line. Its displacement from O is \(s=t^{3} - 12 t^{2} + 36 t\) metres after \(t\) seconds, \(t\ge0\). For each time, say whether the particle is speeding up or slowing down.

1. (a) \(t=1\)
2. (b) \(t=3\)
3. (c) \(t=7\)

A particle moves in a straight line. Its acceleration is \(a=6 t - 4\) m s\(^{-2}\). When \(t=0\) the velocity is 3 m s\(^{-1}\) and the displacement is 1 m.

1. (a) Find an expression for the velocity.
2. (b) Find an expression for the displacement.
3. (c) Find the displacement when \(t=2\).

A cyclist stops pedalling and freewheels along a straight road. Its distance from the start is \(s=60\left(1-e^{-0.2t}\right)\) metres after \(t\) seconds, \(t\ge0\).

1. (a) Find the velocity when \(t=5\). Give your answer to 3 significant figures.
2. (b) Find the time when the velocity is 1 m s\(^{-1}\). Give your answer to 3 significant figures.
3. (c) Find the distance that the cyclist approaches and never passes.

A seat on a Ferris wheel moves round and up and down. Its height above the ground is \(h=11-9\cos(0.5t)\) m after \(t\) seconds, where \(t\ge0\). Use radians.

1. (a) Find the vertical velocity when \(t=3\). Give your answer to 3 significant figures.
2. (b) Find the greatest rate at which the seat rises or falls.
3. (c) Find the first time that the seat is rising at 3 m s\(^{-1}\). Give your answer to 3 significant figures.

A trolley moves along a straight track. Its velocity is \(v=t^{2} - 7 t + 10\) m s\(^{-1}\) for \(0\le t\le6\). Give your answers to 3 significant figures.

1. (a) Find the times when the velocity is zero.
2. (b) Find the displacement from the starting point after 6 seconds.
3. (c) Find the total distance travelled in the 6 seconds.

A particle moves in a straight line. Its displacement from O is \(s=5te^{-0.4t}\) metres after \(t\) seconds, \(t\ge0\). Give your answers to 3 significant figures where needed.

1. (a) Find expressions for the velocity and the acceleration.
2. (b) Find the time when the particle is at rest, and its displacement then.
3. (c) Find the time when the acceleration is zero, and the velocity then.

A particle moves in a straight line. Its velocity is \(v=5\sin(0.8t)-2\) m s\(^{-1}\) for \(0\le t\le8\). Use radians and give your answers to 3 significant figures.

1. (a) Find the times when the particle is at rest.
2. (b) Find the total distance travelled in the 8 seconds.
3. (c) Find the displacement from the starting point after 8 seconds.