Vector motion with components

6 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are \includegraphics[max width=\textwidth, alt={}, center]{63a2dc41-5e8b-4275-8653-ece5067c4306-5_398_689_434_689} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

• During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
• During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
• Alesha's mass, including her equipment, is 100 kg .
• Initially, her position vector is \(\left( \begin{array} { r } - 75 \\ 90 \\ 750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5 \\ 0 \\ - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
  1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of \(\left( \begin{array} { r } 0 \\ 0 \\ - 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0 \\ 0 \\ 880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50 \\ - 20 \\ 0 \end{array} \right) \mathrm { N }\) are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

2 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.

1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } { } ^ { 1 }\) when \(t = 1\).
3. Find an expression for the velocity of the particle at time \(t\).

2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the acceleration of the particle at time \(t\).
2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
3. Calculate the speed of the particle when \(t = 1\).

1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
   2. State the direction in which the particle is travelling at this time.
3. Find the acceleration of the particle when \(t = 4\).
4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).

2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds.
2. Find the speed of \(P\) when \(t = 0.5\).

8 A boat is initially at the origin, heading due east at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then experiences a constant acceleration of \(( - 0.2 \mathbf { i } + 0.25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. State the initial velocity of the boat as a vector.
2. Find an expression for the velocity of the boat \(t\) seconds after it has started to accelerate.
3. Find the value of \(t\) when the boat is travelling due north.
4. Find the bearing of the boat from the origin when the boat is travelling due north.

3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the speed of the particle when \(t = 3\).
   2. State the direction in which the particle is travelling when \(t = 3\).
3. Find the acceleration of the particle when \(t = 3\).
4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).

11 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds is given by \(\mathbf { r } = 2 t ^ { 3 } \mathbf { i } + \left( 5 t ^ { 2 } - 4 t \right) \mathbf { j }\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044 ^ { \circ }\).
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\).
3. Determine the times at which the particle is moving on a bearing of \(045 ^ { \circ }\).

At time \(t\) seconds, \(t \geq 0\), a particle \(P\) has velocity \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (27 - 3t^2)\mathbf{i} + (8 - t^3)\mathbf{j}$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf{r}\) m relative to a fixed origin \(O\), where \(\mathbf{r} = -5\mathbf{i} + 2\mathbf{j}\) Find

1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf{i}\), [5]
2. the position vector of \(P\) at the instant when \(t = 3\) [5]

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

The velocity v m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is 3i m.

1. Find the distance of \(P\) from \(O\) when \(t = 2\). [6]

A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\). Find

1. the speed of \(P\) when \(t = 4\) [2]
2. the acceleration of \(P\) when \(t = 4\) [3]

Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),

1. find the position vector of \(P\) when \(t = 4\) [5]

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$\mathbf{r} = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

The velocity \(v\) m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin O is \(3\mathbf{i}\) m.

1. Find the distance of \(P\) from O when \(t = 2\). [6]

A particle has mass 6 kg. A single force \((24e^{-2t}\mathbf{i} - 12t^3\mathbf{j})\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.

1. Find the acceleration of the particle at time \(t\). [2 marks]
2. At time \(t = 0\), the velocity of the particle is \((-7\mathbf{i} - 4\mathbf{j}) \text{ m s}^{-1}\). Find the velocity of the particle at time \(t\). [4 marks]
3. Find the speed of the particle when \(t = 0.5\). [4 marks]

A particle moves in a horizontal plane under the action of a single force, \(\mathbf{F}\) newtons. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$

1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
2. The mass of the particle is \(2\) kg.
   1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
   2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]

A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf{r}\) m relative to \(O\) after \(t\) seconds is given by \(\mathbf{r} = at^2\mathbf{i} + bt\mathbf{j}\). 60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \((90\mathbf{i} + 30\mathbf{j})\) m.

1. Find the values of the constants \(a\) and \(b\). [3 marks]
2. Find the speed of \(P\) when it is at \(Q\). [4 marks]
3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\). [2 marks]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \((2t + 4)\mathbf{i} + (kt^2 - 5)\mathbf{j}\).

1. Find the value of \(k\) if \(P\) takes \(4\) seconds to reach \(Y\). [3 marks]
2. Show that \(P\) has constant acceleration and find the magnitude and direction of this acceleration. [4 marks]

A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf{r} = t^2\mathbf{i} - 2t\mathbf{j}\).

1. Find the velocity vector of \(P\) at time \(t\) seconds. [2 marks]
2. Show that the direction of the acceleration of \(P\) is constant. [2 marks]
3. Find the value of \(t\) when the acceleration of \(P\) has magnitude 12 ms\(^{-2}\). [3 marks]

At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]

A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)

A particle \(P\) of mass \(0.5\) kg moves on a horizontal plane such that its velocity vector \(\mathbf{v}\) ms\(^{-1}\) at time \(t\) seconds is given by $$\mathbf{v} = 12\cos(3t)\mathbf{i} - 5\sin(2t)\mathbf{j}.$$

1. Find an expression for the force acting on \(P\) at time \(t\) s. [3]
2. Given that when \(t = 0\), \(P\) has position vector \((\mathbf{4i} + \mathbf{7j})\) m relative to the origin \(O\), find an expression for the position vector of \(P\) at time \(t\) s. [4]
3. Hence determine the distance of \(P\) from \(O\) at time \(t = \frac{\pi}{2}\). [2]

The position vector \(\mathbf{x}\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by $$\mathbf{x} = 3\sin t \mathbf{i} - 4\cos 2t \mathbf{j} + 5\sin t \mathbf{k}.$$

1. Find an expression for the velocity vector \(\mathbf{v}\text{ ms}^{-1}\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest. [7]
2. Write down the momentum vector at time \(t\) seconds. [1]
3. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds. [3]

At time \(t = 0\) seconds, a particle \(A\) has position vector \((6\mathbf{i} + 2\mathbf{j} - 8\mathbf{k})\) m relative to a fixed origin \(O\) and is moving with constant velocity \((3\mathbf{i} - \mathbf{j} + 4\mathbf{k})\) ms\(^{-1}\).

1. Write down the position vector of particle \(A\) at time \(t\) seconds and hence find the distance \(OA\) when \(t = 5\). [4]
2. The position vector, \(\mathbf{r}_B\) metres, of another particle \(B\) at time \(t\) seconds is given by $$\mathbf{r}_B = 3\sin\left(\frac{t}{2}\right)\mathbf{i} - 3\cos\left(\frac{t}{2}\right)\mathbf{j} + 5\mathbf{k}.$$
   1. Show that \(B\) is moving with constant speed.
   2. Determine the smallest value of \(t\) such that particles \(A\) and \(B\) are moving perpendicular to each other. [7]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by \(\mathbf{r} = 2t^2\mathbf{i} + (5t^2 - 4t)\mathbf{j}\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044°\). [3]
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\). [4]
3. Determine the times at which the particle is moving on a bearing of \(045°\). [2]