Kinematics on a line Problem set B

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) \(s = 5\sin 3t\)
2. (b) \(s = 2e^{-4t}\)
3. (c) \(s = (5t - 2)^{3}\)

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) \(v = 2s + 5\)
2. (b) \(v = s^{2} - 6\)
3. (c) \(v = e^{2s}\)

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

1. (a) \(\displaystyle\int_0^4 8\cos(0.5t)\,dt\) (change in velocity, \(a = 8\cos(0.5t)\))
2. (b) \(\displaystyle\int_0^{1.5} 6te^{-2t}\,dt\) (change in displacement, \(v = 6te^{-2t}\))

A mass hangs on a spring and moves up and down. Its displacement from the rest position is \(s = 0.12\cos 8t\) metres at time \(t\) seconds.

1. (a) Find the velocity when \(t = 0.25\).
2. (b) Find the acceleration when \(t = 0.25\).

A boat drifts towards a pier with the engine in reverse. Its velocity is \(v = 18 - 2s\) m s⁻¹ when it is \(s\) metres from the point where the engine was reversed.

1. (a) Find an expression for the acceleration in terms of \(s\).
2. (b) Find the acceleration when \(s = 5\). What does the sign tell you?

A skater pushes off a wall and her acceleration is \(a = 7e^{-0.5t}\) m s⁻², with \(t\) in seconds.

1. (a) Find the change in velocity between \(t = 1\) and \(t = 5\).
2. (b) The velocity is 8 m s⁻¹ when \(t = 1\). Find the velocity when \(t = 5\).

A sensor on a vibrating plate moves on a line with displacement \(s = 3\cos 4t + 2\) millimetres.

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

A bead slides on a wire. Its velocity satisfies \(v^2 = 25 - 4s^2\). Use \(a = \dfrac{d}{ds}\left(\tfrac12 v^2\right)\) to find an expression for the acceleration in terms of \(s\).