Particle motion: 2D constant acceleration

4 A particle is initially at the origin, moving with velocity \(\mathbf { u }\). Its acceleration \(\mathbf { a }\) is constant. At time \(t\) its displacement from the origin is \(\mathbf { r } = \binom { x } { y }\), where \(\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }\).

1. Write down \(\mathbf { u }\) and \(\mathbf { a }\) as column vectors.
2. Find the speed of the particle when \(t = 2\).
3. Show that the equation of the path of the particle is \(y = 3 x - x ^ { 2 }\).

6 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors east and north respectively. Position vectors are with respect to an origin O . Time \(t\) is in seconds. A skater has a constant acceleration of \(- 2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At \(t = 0\), his velocity is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his position vector is \(3 \mathbf { j } \mathrm {~m}\).

1. Find expressions in terms of \(t\) for the velocity and the position vector of the skater at time \(t\).
2. Calculate as a bearing the direction of motion of the skater when \(t = 2.5\).

5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. 1. Find the velocity of the particle when \(t = 22.5\).
   2. State the direction in which the particle is travelling at this time.
3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 A helicopter is initially at rest on the ground at the origin when it begins to accelerate in a vertical plane. Its acceleration is \(( 4.2 \mathbf { i } + 2.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for the first 20 seconds of its motion. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. Assume that the helicopter moves over horizontal ground.

1. Find the height of the helicopter above the ground at the end of the 20 seconds.
2. Find the velocity of the helicopter at the end of the 20 seconds.
3. Find the speed of the helicopter when it is at a height of 180 metres above the ground.

7 A particle moves on a smooth horizontal plane. It is initially at the point \(A\), with position vector \(( 9 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\), and has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves with a constant acceleration of \(( 0.25 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for 20 seconds until it reaches the point \(B\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the velocity of the particle at the point \(B\).
2. Find the velocity of the particle when it is travelling due north.
3. Find the position vector of the point \(B\).
4. Find the average velocity of the particle as it moves from \(A\) to \(B\).
   
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-15_2484_1709_223_153}

7 A helicopter is initially hovering above a lighthouse. It then sets off so that its acceleration is \(( 0.5 \mathbf { i } + 0.375 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The helicopter does not change its height above sea level as it moves. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the speed of the helicopter 20 seconds after it leaves its position above the lighthouse.
2. Find the bearing on which the helicopter is travelling, giving your answer to the nearest degree.
3. The helicopter stops accelerating when it is 500 metres from its initial position. Find the time that it takes for the helicopter to travel from its initial position to the point where it stops accelerating.

7 A particle moves with a constant acceleration of \(( 0.1 \mathbf { i } - 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). It is initially at the origin where it has velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the position vector of the particle \(t\) seconds after it has left the origin.
2. Find the time that it takes for the particle to reach the point where it is due east of the origin.
3. Find the speed of the particle when it is travelling south-east.

8 A helicopter travels at a constant height above the sea. It passes directly over a lighthouse with position vector \(( 500 \mathbf { i } + 200 \mathbf { j } )\) metres relative to the origin, with a velocity of \(( - 17.5 \mathbf { i } - 27 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The helicopter moves with a constant acceleration of \(( 0.5 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the position vector of the helicopter \(t\) seconds after it has passed over the lighthouse.
2. The position vector of a rock is \(( 200 \mathbf { i } - 400 \mathbf { j } )\) metres relative to the origin. Show that the helicopter passes directly over the rock, and state the time that it takes for the helicopter to move from the lighthouse to the rock.
3. Find the average velocity of the helicopter as it moves from the lighthouse to the rock.
4. Is the magnitude of the average velocity equal to the average speed of the helicopter? Give a reason for your answer.

4 A particle moves with constant acceleration between the points \(A\) and \(B\). At \(A\), it has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(B\), it has velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It takes 10 seconds to move from \(A\) to \(B\).

1. Find the acceleration of the particle.
2. Find the distance between \(A\) and \(B\).
3. Find the average velocity as the particle moves from \(A\) to \(B\).

2. A ball of mass 2 kg moves on a smooth horizontal surface under the action of a constant force, \(\mathbf { F }\). The initial velocity of the ball is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and 4 seconds later it has velocity \(( 10 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors.

1. Making reference to the mass of the ball and the force it experiences, explain why it is reasonable to assume that the acceleration is constant.
2. Find, giving your answers correct to 3 significant figures,
   1. the magnitude of the acceleration experienced by the ball,
   2. the angle which \(\mathbf { F }\) makes with the vector \(\mathbf { i }\).

3 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
east, north and vertically upwards. \includegraphics[max width=\textwidth, alt={}, center]{cb72a1c4-f769-4348-ad7f-66c3c96e1732-3_401_686_368_721} 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.

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 2\mathbf{j}) \text{ms}^{-2}\). At time \(t = 4\) seconds, \(P\) has velocity \(6\mathbf{i} \text{ms}^{-1}\). Determine the speed of \(P\) at time \(t = 0\) seconds. [4]

A particle of mass 0.1 kg is initially stationary. A single force \(\mathbf{F}\) acts on this particle in a direction parallel to the vector \(7\mathbf{i} + 24\mathbf{j}\) As a result, the particle accelerates in a straight line, reaching a speed of \(4\text{ m s}^{-1}\) after travelling a distance of 3.2 m Find \(\mathbf{F}\). [5 marks]

A particle P lies at rest on a smooth horizontal table. A constant resultant force, F newtons, is then applied to P. As a result P moves in a straight line with constant acceleration \(\begin{bmatrix}8\\6\end{bmatrix}\) m s⁻²

1. Show that the magnitude of the acceleration of P is 10 m s⁻² [1 mark]
2. Find the speed of P after 3 seconds. [1 mark]
3. Given that \(\mathbf{F} = \begin{bmatrix}2\\1.5\end{bmatrix}\) N, find the mass of P. [2 marks]