Projectiles and motion with variable velocity Problem set B

A jet of water leaves a nozzle at ground level. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 11 \\ 24 - 9.8t \end{pmatrix}\) m s⁻¹, where the first component is horizontal and the second is upward. It starts at the origin.

1. (a) Find the time at which it reaches its greatest height.
2. (b) Find its displacement from the origin at time \(t\).
3. (c) Find its greatest height.

A firework shell is launched from level ground. Its displacement from the launch point at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix} 7t \\ 16t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. The ground is level.

1. (a) Find the time at which it lands.
2. (b) Find how far from the origin it lands.

A kite drifts in a vertical plane. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 3t + 4 \\ 18 - 0.3t^2 \end{pmatrix}\) m s⁻¹. Find, when \(t = 4\):

1. (a) the angle that the velocity makes with the direction of \(\mathbf{i}\)
2. (b) the speed
3. (c) the magnitude of the acceleration

A shuttle moves along a track in a factory. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 6 + 2t \\ 9 - 5t \end{pmatrix}\) m s⁻¹.

1. (a) Find the speed when \(t = 2\).
2. (b) Show that the acceleration is constant and find its magnitude.

A basketball is thrown from the centre of a court. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 6 \\ 10 - 9.8t \end{pmatrix}\) m s⁻¹, where the first component is horizontal and the second is upward.

1. (a) State the magnitude and direction of the acceleration.
2. (b) Find the initial speed.
3. (c) Find an expression for the speed at time \(t\), and use technology to find its minimum value.

A rugby ball is kicked on a pitch. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 5 \\ 13 - 9.8t \end{pmatrix}\) m s⁻¹, where the first component is horizontal and the second is upward. It starts at the origin on level ground.

1. (a) Find its displacement from the origin at time \(t\).
2. (b) Show that it lands approximately 2.65 seconds after it is projected, and find its distance from the origin at that time.
3. (c) Show that it lands with the same speed as it was projected with.

A probe is released from rest at the origin in space. Its constant acceleration is \(\mathbf{a} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) m s⁻².

1. (a) Find expressions for its velocity and displacement at time \(t\).
2. (b) How long does it take for its speed to reach 30 m s⁻¹?
3. (c) Plot its position in the \(x\)-\(y\) plane when \(t = 0, 1, 2, 3\). Describe the path.

A cyclist rides over a flat car park. Its constant acceleration is \(\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\) m s⁻². When \(t = 0\) it passes through the origin with velocity \(\begin{pmatrix} -1 \\ 6 \end{pmatrix}\) m s⁻¹.

1. (a) Find the velocity and displacement vectors at time \(t\).
2. (b) Find the angle between the initial velocity and the velocity after 3 seconds.
3. (c) Plot the positions at \(t = 0, 1, 2, 3\). Does it move in a straight line?

A rover's ball on a distant planet is launched from the origin with initial velocity \(\begin{pmatrix} 9 \\ 5 \end{pmatrix}\) m s⁻¹, where the first component is horizontal and the second is upward. The greatest height reached is 1.6 m. Find, to two significant figures, the gravitational acceleration \(g\) there.

A robot vacuum cleaner moves around a room. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 2t - 8 \\ 5 - t \end{pmatrix}\) m s⁻¹ and it starts at the origin. Show that it never returns to its starting position.