3.02g Two-dimensional variable acceleration

7 \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817} A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.

1. Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772} A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
3. (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
   (b) Find the magnitude of the impulse exerted by the wall on the sphere.
4. Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.

11 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_384_524_587_808} One end of a light inextensible string of length 10 m is attached to a fixed point \(O\). A particle of mass 5 kg is attached to the other end of the string. The particle rests in equilibrium below \(O\). The particle is pulled aside until the string makes an angle of \(60 ^ { \circ }\) with the downward vertical and released from rest (see diagram). At the instant when the string makes an angle \(\cos ^ { - 1 } \left( \frac { 4 } { 5 } \right)\) with the downward vertical,

1. find the speed of the particle and the tension in the string,
2. show that the magnitude of the acceleration of the particle is \(6 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$

1. Show that the distance \(OP\) is 6.25 m. [5]

On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).

1. Find the values of the constants \(k\) and \(c\). [5]

1. Find the acceleration of the particle at time \(t = 5\). [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 \(t\) s after being dropped, the acceleration, \(a\text{ m s}^{-2}\), of the box in the vertically downwards direction is modelled by $$a = 10 - t \text{ for } 0 \leqslant t \leqslant 10,$$ $$a = 0 \text{ for } t > 10.$$

1. Find an expression for the velocity, \(v\text{ m s}^{-1}\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10\), \(v = 50\). [4]
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\). [3]
3. Show that the height, \(h\) m, of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5t^2 + \frac{1}{6}t^3.$$ Find the height of the box when \(t = 10\). [4]
4. Find the value of \(t\) when the box hits the ground. [2]
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.
   1. The box should be dropped from a height of 500 m instead of 1000 m. [2]
   2. The box should be fitted with a parachute so that its acceleration is given by $$a = 10 - 2t \text{ for } 0 \leqslant t \leqslant 5,$$ $$a = 0 \text{ for } t > 5.$$ [3]

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 \(P\) 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]

A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x\) m towards \(A\). The particle \(P\) is next at rest when \(t = 0.25\pi\) having travelled a distance of \(1.2\) m.

1. Find the maximum velocity of \(P\). [3]
2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\). [4]
3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\)'s speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\). [4]

An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, \(v \text{ m s}^{-1}\), may be modelled by $$v = 11.71 - 11.68e^{-0.9t} - 0.03e^{0.3t}$$ where \(t\) is the time, in seconds, after the starting pistol is fired.

1. Find the maximum value of \(v\), giving your answer to one decimal place. Fully justify your answer. [8 marks]
2. Find an expression for the distance run in terms of \(t\). [6 marks]
3. The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model. [2 marks]

At time \(t\) seconds, where \(t \geq 0\), a particle P has position vector \(\mathbf{r}\) metres, where $$\mathbf{r} = (2t^2 - 12t + 6)\mathbf{i} + (t^3 + 3t^2 - 8t)\mathbf{j}.$$ The velocity of P at time \(t\) seconds is \(v \text{ m s}^{-1}\).

1. Find \(v\) in terms of \(t\). [1]
2. Determine the speed of P at the instant when it is moving parallel to the vector \(\mathbf{i} - 4\mathbf{j}\). [5]
3. Determine the value of \(t\) when the magnitude of the acceleration of P is \(20.2 \text{ m s}^{-2}\). [3]

In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the \(x\)- and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leq t \leq 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O. The acceleration of C is given by \(2\mathbf{i} + t\mathbf{j} \text{ cm s}^{-2}\). When \(t = 4\), the displacement of C from O is \(16\mathbf{i} + \frac{32}{3}\mathbf{j}\) cm, and the velocity of C is \(8\mathbf{i} \text{ cm s}^{-1}\). Determine a cartesian equation for the path of C for \(0 < t < 5\). You are not required to simplify your answer. [6]

A particle P has position vector \(\mathbf{r}\) m at time \(t\) s given by \(\mathbf{r} = (t^3 - 3t^2)\mathbf{i} - (4t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Find the magnitude of the acceleration of P when \(t = 2\). [4]

A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the magnitude of the acceleration of the particle. [3]
2. Given that at time \(t = 0\) seconds, the position vector of the particle is \(2\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}\) and it is moving with velocity \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), find the position vector of the particle when \(t = 2\) seconds. [3]

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]

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]