Projectiles, circular motion and delayed objects Problem set B

A jet of water leaves a nozzle at the origin. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}8\\14.7 − 9.8t\end{pmatrix}\) m s⁻¹. It starts at the origin.

1. (a) Find the angle that the velocity makes with the horizontal (i) at the start (ii) when \(t = 2\).
2. (b) Find the time at which it reaches its greatest height.
3. (c) Find an expression for the displacement from the origin at time \(t\). Hence find the greatest height.

A javelin is thrown. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}9\\17.64 − 9.8t\end{pmatrix}\) m s⁻¹, where the origin is the launch point on level ground. Find the distance from the origin to the point where it lands on the same level ground.

A toy train runs on a circular track. Its position vector relative to the origin at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}1.8\cos 5t\\1.8\sin 5t\end{pmatrix}\) m.

1. (a) By finding \(|\mathbf{r}|\), show that the object moves in a circle. State the radius.
2. (b) Find the velocity vector.
3. (c) Show that the velocity is always perpendicular to the position vector.
4. (d) Show that the speed is constant and find its value.

Two objects move in circles about the origin. Find the speed of each, in m s⁻¹.

1. (a) A bicycle valve: \(\mathbf{v} = \begin{pmatrix}-0.9\sin 2t\\0.9\cos 2t\end{pmatrix}\)
2. (b) A ride car: \(\mathbf{v} = \begin{pmatrix}5\cos 0.2t\\5\sin 0.2t\end{pmatrix}\)

A mirror on a rotating platform moves in a circle. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}4\cos 2t\\4\sin 2t\end{pmatrix}\) m.

1. (a) Find the velocity and then the acceleration.
2. (b) Show that \(\mathbf{a} = k\mathbf{r}\) and find \(k\).
3. (c) State the direction of the acceleration.
4. (d) Show that the magnitude of the acceleration is constant and find its value.

A lamp on a lighthouse arm moves in a circle. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}2.1\cos 0.7t\\2.1\sin 0.7t\end{pmatrix}\) m s⁻¹. At \(t = 0\) the position vector is \(\begin{pmatrix}0\\−3\end{pmatrix}\) m.

1. (a) Find the position vector at time \(t\), show that the object moves in a circle about the origin and state the radius.
2. (b) Write down the initial position and the radius of the circle.
3. (c) Find the time \(T\) when the object first returns to its starting position.
4. (d) Verify that the speed equals \(\dfrac{2\pi r}{T}\).

A model boat moves on a pond. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}4 − 3t\\1 + 2t^2\end{pmatrix}\) m s⁻¹. It starts at the origin.

1. (a) Find an expression for the displacement from the origin at time \(t\).
2. (b) Find the distance from the origin when \(t = 3\).

A skater starts at the origin when \(t = 0\). Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}2 + 1.2t\\8 − 0.3t^2\end{pmatrix}\) m s⁻¹. A second skater follows the same path with the same velocity but leaves the origin 3 seconds later. Find the distance between the two skaters 2 seconds after the second one leaves.

A hose sprays a jet of water. The velocity of the first at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}15\\19.6 − 9.8t\end{pmatrix}\) m s⁻¹. A second hose sprays a second jet. The velocity of the second is \(\mathbf{v} = \begin{pmatrix}20\\b − 9.8t\end{pmatrix}\) m s⁻¹. Both start at the origin and land on level ground in the same place. Find the value of \(b\).

A stone is thrown from the origin. The acceleration is \(\begin{pmatrix}0\\−9.8\end{pmatrix}\) m s⁻² and the initial velocity is \(\begin{pmatrix}5\\16\end{pmatrix}\) m s⁻¹.

1. (a) Find the displacement from the starting point at time \(t\) seconds after the launch.
2. (b) A second identical launch is made 3 seconds after the first, from the same point. Find the displacement of the second object from the starting point at time \(t\) seconds after the first launch, for \(t \geq 3\).

Two rescue flares are fired. The first is launched from the origin at time \(t = 0\) with initial velocity \(\begin{pmatrix}6\\24\end{pmatrix}\) m s⁻¹. The second is launched from the origin 1 second later with initial velocity \(\begin{pmatrix}9\\b\end{pmatrix}\) m s⁻¹. Both move with acceleration \(\begin{pmatrix}0\\−9.8\end{pmatrix}\) m s⁻². The two collide.

1. (a) Find the value of \(b\).
2. (b) Find the distance from the origin of the point where they collide.

A pellet is launched from the origin with initial velocity \(\begin{pmatrix}24\\40\end{pmatrix}\) m s⁻¹ and moves with acceleration \(\begin{pmatrix}0\\−9.8\end{pmatrix}\) m s⁻². A second pellet is launched from the origin with the same initial velocity 3 seconds later.

1. (a) Find an expression for the distance between the two pellets \(t\) seconds after the first launch, for \(t \geq 3\).
2. (b) Use your calculator to find the time, after the first launch, at which the distance is least while both are in flight.

The tip of a turbine blade moves in a circle about the origin. Its position vector is \(\mathbf{r} = \begin{pmatrix}a\cos n t\\a\sin n t\end{pmatrix}\), where \(a\) and \(n\) are positive constants.

1. (a) Show that it moves in a circle and state its radius.
2. (b) Show that its speed is \(v = an\).
3. (c) Show that the time for one full circle is \(T = \dfrac{2\pi}{n}\).
4. (d) Show that the magnitude of the acceleration is \(\dfrac{v^2}{a}\).
5. (e) For \(a = 0.8\) m and \(n = 4\) rad s⁻¹, find the speed, the time for one full circle and the magnitude of the acceleration.

A satellite is modelled as moving in a circle about the origin. Its position vector is \(\mathbf{r} = \begin{pmatrix}R\cos k t\\R\sin k t\end{pmatrix}\), where \(R\) and \(k\) are positive constants.

1. (a) Show that the velocity is perpendicular to the position vector.
2. (b) Show that the acceleration is directed towards the centre of the circle.