Kinematics: displacement, velocity and acceleration Problem set B

A bead slides along a straight wire. The displacement from a fixed point is \(s\) cm, where \(s=3 t^{3} - 10 t^{2} + 7 t + 2\) and \(t\ge0\) is in seconds.

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

A raft drifts along a straight river. The displacement from a fixed point is \(s=20e^{-0.5t}+2t\) m, where \(t\ge0\) is in seconds. Give your answers to 3 significant figures.

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

A sled slides along a straight slope. The displacement from a fixed point is \(s=2 t^{3} - 21 t^{2} + 60 t\) 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 weight bobs on an elastic string. The displacement from its rest position is \(x=5\cos(2t)\) cm, 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 cyclist rides along a straight road. Its displacement from O is \(s=t^{3} - 15 t^{2} + 63 t\) metres after \(t\) seconds, \(t\ge0\). For each time, say whether the particle is speeding up or slowing down.

1. (a) \(t=2\)
2. (b) \(t=4\)
3. (c) \(t=6\)

A skater glides along a straight line of ice. Its acceleration is \(a=6 t - 10\) m s\(^{-2}\). When \(t=0\) the velocity is 7 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 \(t=3\).

A sledge slides along a flat snowfield after a push. Its distance from the start is \(s=45\left(1-e^{-0.15t}\right)\) metres after \(t\) seconds, \(t\ge0\).

1. (a) Find the velocity when \(t=4\). Give your answer to 3 significant figures.
2. (b) Find the time when the velocity is 0.5 m s\(^{-1}\). Give your answer to 3 significant figures.
3. (c) Find the distance that the sledge approaches and never passes.

The top of an engine piston moves up and down. Its height above the ground is \(h=15-6\cos(1.2t)\) cm after \(t\) seconds, where \(t\ge0\). Use radians.

1. (a) Find the vertical velocity when \(t=2\). Give your answer to 3 significant figures.
2. (b) Find the greatest rate at which the piston rises or falls.
3. (c) Find the first time that the piston is rising at 4 cm s\(^{-1}\). Give your answer to 3 significant figures.

A train moves along a straight section of line. Its velocity is \(v=t^{2} - 8 t + 12\) m s\(^{-1}\) for \(0\le t\le7\). 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 7 seconds.
3. (c) Find the total distance travelled in the 7 seconds.

A probe moves along a straight tube. Its displacement from O is \(s=8te^{-0.5t}\) 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 drone moves along a straight line. Its velocity is \(v=6\sin(0.5t)-2.5\) m s\(^{-1}\) for \(0\le t\le9\). 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 9 seconds.
3. (c) Find the displacement from the starting point after 9 seconds.