3.02g Two-dimensional variable acceleration

6 A child pulls a sledge, of mass 8 kg , along a rough horizontal surface, using a light rope. The coefficient of friction between the sledge and the surface is 0.3 . The tension in the rope is \(T\) newtons. The rope is kept at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-4_273_775_516_644} Model the sledge as a particle.

1. Draw a diagram to show all the forces acting on the sledge.
2. Find the magnitude of the normal reaction force acting on the sledge, in terms of \(T\).
3. Given that the sledge accelerates at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find \(T\).

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.

7 Two particles, \(A\) and \(B\), move on a horizontal surface with constant accelerations of \(- 0.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. At time \(t = 0\), particle \(A\) starts at the origin with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\), particle \(B\) starts at the point with position vector \(11.2 \mathbf { i }\) metres, with velocity \(( 0.4 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the position vector of \(A , 10\) seconds after it leaves the origin.
   [0pt] [2 marks]
2. Show that the two particles collide, and find the position vector of the point where they collide.
   [0pt] [9 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-16_1881_1707_822_153}
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-17_2484_1707_221_153}

2 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10 \\ & a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$

1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
4. Find the value of \(t\) when the box hits the ground.
5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
   (A) The box should be dropped from a height of 500 m instead of 1000 m .
   (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$

2 \begin{figure}[h] \end{figure} A toy car is moving along the straight line \(\mathrm { O } x\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at \(\mathrm { A } , 3 \mathrm {~m}\) from O as shown in Fig. 5. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 2 + 12 t - 3 t ^ { 2 }$$ Calculate the distance of the car from O when its acceleration is zero.

4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),

1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)

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

1. Find the magnitude of the acceleration of \(P\) when \(t = 2\). When \(t = 0\), the displacement of \(P\) from a fixed origin \(O\) is \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
2. Show that the displacement of \(P\) from \(O\) when \(t = 6\) is given by \(k ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(k\) is an integer which you should find.
   (6 marks)

2. A particle \(P\) moves along the \(x\)-axis such that its displacement, \(x\) metres, from the origin \(O\) at time \(t\) seconds is given by $$x = 2 + t - \frac { 1 } { 10 } \mathrm { e } ^ { t }$$

1. Find the distance of \(P\) from \(O\) when \(t = 0\).
2. Find, correct to 1 decimal place, the value of \(t\) when the velocity of \(P\) is zero.
   (4 marks)

3 A particle \(P\) of mass 0.2 kg moves on a smooth horizontal plane. Initially it is projected with velocity \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) towards another fixed point \(A\). At time \(t\) s after projection, \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the direction \(O A\) being positive. A force of \(( 1.5 t - 1 ) \mathrm { N }\) acts on \(P\) in the direction parallel to \(O A\).

1. Find an expression for \(v\) in terms of \(t\).
2. Find the time when the velocity of \(P\) is next \(0.8 \mathrm {~ms} ^ { - 1 }\).
3. Find the times when \(P\) subsequently passes through \(O\).
4. Find the distance \(P\) travels in the third second of its motion.

4 A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving away from \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of magnitude \(1.6 v ^ { 2 } \mathrm {~N}\) resisting the motion of \(P\).

1. Find an expression for \(\frac { \mathrm { d } v } { \mathrm {~d} x }\) in terms of \(v\), and hence show that \(v = 2 \mathrm { e } ^ { - 4 x }\).
2. Find the distance travelled by \(P\) in the 0.5 seconds after it leaves \(O\).

3 An aircraft of mass 80000 kg travelling at \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) touches down on a straight horizontal runway. It is brought to rest by braking and resistive forces which together are modelled by a horizontal force of magnitude ( \(27000 + 50 v ^ { 2 }\) ) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the aircraft. Find the distance travelled by the aircraft between touching down and coming to rest.

6. \begin{figure}[h] \end{figure} A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time \(t = 0\), a boat leaves a point \(O\) on one bank and moves in a straight line across the river to a point \(P\) on the opposite bank. Its path \(O P\) is perpendicular to both banks and \(O P = 30 \mathrm {~m}\), as shown in Figure 2. The speed of flow of the river, \(r \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a point on \(O P\) which is at a distance \(x \mathrm {~m}\) from \(O\), is modelled as $$r = \frac { 1 } { 10 } x , \quad 0 \leq x \leq 30$$ The speed of the boat relative to the water is constant at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the boat is at a distance \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O P\).

1. Show that $$100 v ^ { 2 } = 2500 - x ^ { 2 }$$
2. Hence show that $$\frac { \mathbf { d } ^ { 2 } x } { \mathbf { d } t ^ { 2 } } + \frac { x } { 100 } = 0$$
3. Find the total time taken for the boat to cross the river from \(O\) to \(P\).
   (9)

5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).

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 \(t = 0.5\).

5 The constant forces \(\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )\) newtons and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )\) newtons act on a particle. No other forces act on the particle.

1. Find the resultant force acting on the particle.
2. Given that the mass of the particle is 4 kg , show that the acceleration of the particle is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
3. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. When \(t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }\). Show that \(\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }\) when \(t = 0\).
   2. Write down an expression for \(\mathbf { v }\) at time \(t\).
   3. Find the times when the speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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.

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 \(045 ^ { \circ }\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to the fixed point \(O\).]

A particle \(P\) moves with constant acceleration.
At time \(t = 0\), the particle is at \(O\) and is moving with velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) At time \(t = 2\) seconds, \(P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m.

1. Show that the magnitude of the acceleration of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At the instant when \(P\) leaves the point \(A\), the acceleration of \(P\) changes so that \(P\) now moves with constant acceleration ( \(4 \mathbf { i } + 8.8 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 2 }\) At the instant when \(P\) reaches the point \(B\), the direction of motion of \(P\) is north east.
2. Find the time it takes for \(P\) to travel from \(A\) to \(B\).

6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { m } \mathrm { s } ^ { - 2 }\) is given by $$\mathbf { a } = 5 t \mathbf { i } - 15 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$ When \(t = 0\), the velocity of \(P\) is \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the speed of \(P\) when \(t = 4\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively]

A radio controlled model boat is placed on the surface of a large pond.
The boat is modelled as a particle.
At time \(t = 0\), the boat is at the fixed point \(O\) and is moving due north with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Relative to \(O\), the position vector of the boat at time \(t\) seconds is \(\mathbf { r }\) metres.
At time \(t = 15\), the velocity of the boat is \(( 10.5 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
The acceleration of the boat is constant.

1. Show that the acceleration of the boat is \(( 0.7 \mathbf { i } - 0.1 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find \(\mathbf { r }\) in terms of \(t\).
3. Find the value of \(t\) when the boat is north-east of \(O\).
4. Find the value of \(t\) when the boat is moving in a north-east direction.

6 A particle, of mass 5 kg , moves on a circular path so that at time \(t\) seconds it has position vector \(\mathbf { r }\) metres, where $$\mathbf { r } = ( 2 \sin 3 t ) \mathbf { i } + ( 2 \cos 3 t ) \mathbf { j }$$ 6

1. Prove that the velocity of the particle is perpendicular to its position vector.
   6
2. Prove that the magnitude of the resultant force on the particle is constant.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) is moving with velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point \(O\) At time \(t = 0 , P\) is at the point with position vector \(( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }\).

1. Find the position vector of \(P\) when \(t = 3\)
2. Find the magnitude of the acceleration of \(P\) when \(t = 3\) At time \(T\) seconds, \(P\) is moving in the direction of the vector \(2 \mathbf { i } + \mathbf { j }\)
3. Find the value of \(T\)