Kinematics on a line Problem set A

The displacement \(s\) metres of an object from a fixed point at time \(t\) seconds is given. Find expressions for the velocity and acceleration.

1. (a) \(s = 4\cos 5t\)
2. (b) \(s = 3e^{-2t}\)
3. (c) \(s = (2t + 1)^{4}\)

A particle moves on a line. Its velocity \(v\) is given in terms of its displacement \(s\). Find an expression for the acceleration in terms of \(s\).

1. (a) \(v = 3s + 2\)
2. (b) \(v = 4 - s^{2}\)
3. (c) \(v = e^{s}\)

Use your calculator to evaluate each definite integral. Give each answer to 3 significant figures. Time is in seconds.

1. (a) \(\displaystyle\int_1^3 5\sin 2t\,dt\) (change in velocity, \(a = 5\sin 2t\))
2. (b) \(\displaystyle\int_0^2 3te^{-t}\,dt\) (change in displacement, \(v = 3te^{-t}\))

A piston rod moves in a straight line. Its displacement from the centre of the cylinder is \(s = 0.04\sin 50t\) metres at time \(t\) seconds.

1. (a) Find the velocity when \(t = 0.2\).
2. (b) Find the acceleration when \(t = 0.2\).

A cyclist brakes as she approaches a junction. Her velocity is \(v = 24 - 3s\) m s⁻¹ when she is \(s\) metres past the point where she starts braking.

1. (a) Find an expression for the acceleration in terms of \(s\).
2. (b) Find the acceleration when \(s = 4\). What does the sign tell you?

A rocket sled's acceleration is \(a = 9e^{-0.3t}\) m s⁻², with \(t\) in seconds.

1. (a) Find the change in velocity between \(t = 2\) and \(t = 6\).
2. (b) The velocity is 15 m s⁻¹ when \(t = 2\). Find the velocity when \(t = 6\).

A particle moves on a line with displacement \(s = 5\sin 2t + 1\) metres.

1. (a) Find an expression for \(a\) in terms of \(t\).
2. (b) Find the constant \(k\) such that \(a = k(s - 1)\).

A particle moves on a line. Its velocity satisfies \(v^2 = 16 - s^2\). Use \(a = \dfrac{d}{ds}\left(\tfrac12 v^2\right)\) to find an expression for the acceleration in terms of \(s\).