Acceleration as function of position

5 The acceleration of a particle moving in a straight line is \(( x - 2.4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) when its displacement from a fixed point \(O\) of the line is \(x \mathrm {~m}\). The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it is given that \(v = 2.5\) when \(x = 0\). Find

1. an expression for \(v\) in terms of \(x\),
2. the minimum value of \(v\).

2 A particle starts from rest at \(O\) and travels in a straight line. Its acceleration is \(( 3 - 2 x ) \mathrm { ms } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of the particle from \(O\).

1. Find the value of \(x\) for which the velocity of the particle reaches its maximum value.
2. Find this maximum velocity.

4 A particle \(P\) starts from rest at a point \(O\) and travels in a straight line. The acceleration of \(P\) is \(( 15 - 6 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
2. Calculate the acceleration of \(P\) when it is at instantaneous rest and \(x > 0\).

5 A particle \(P\) is moving along a straight line with acceleration \(3 \mathrm { ku } - \mathrm { kv }\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.

1. Find the time taken for \(P\) to achieve a velocity of \(2 u\).
2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2 u\).

3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\).
\includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
2. Find the set of possible values of \(h\), in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).

3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\).
\includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
2. Find the set of possible values of \(h\), in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).

6 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(6 v \sqrt { v + 9 }\). When \(t = 0 , x = 2\) and \(v = 72\).

1. Find an expression for \(v\) in terms of \(x\).
2. Find an expression for \(x\) in terms of \(t\).

6 A particle of mass \(m \mathrm {~kg}\) falls vertically under gravity, from rest. At time \(t \mathrm {~s} , P\) has fallen \(x \mathrm {~m}\) and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(k m g v \mathrm {~N}\), where \(k\) is a constant.

1. Find an expression for \(v\) in terms of \(t , g\) and \(k\).
2. Given that \(k = 0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12 \mathrm {~ms} ^ { - 1 }\).

6 A particle \(P\) of mass 2 kg moving on a horizontal straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The only horizontal force acting on \(P\) has magnitude \(\frac { 1 } { 10 } ( 2 \mathrm { v } - 1 ) ^ { 2 } \mathrm { e } ^ { - \mathrm { t } } \mathrm { N }\) and acts towards \(O\). When \(t = 0 , x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{b57762bf-7a4f-486d-b9f2-8ae727bfb630-12_69_1569_466_328}
2. Find an expression for \(x\) in terms of \(t\).

6 A particle \(P\) of mass 2 kg moving on a horizontal straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The only horizontal force acting on \(P\) has magnitude \(\frac { 1 } { 10 } ( 2 \mathrm { v } - 1 ) ^ { 2 } \mathrm { e } ^ { - \mathrm { t } } \mathrm { N }\) and acts towards \(O\). When \(t = 0 , x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{c1a3340d-158d-4c37-9577-96074e59ac3d-12_69_1569_466_328}
2. Find an expression for \(x\) in terms of \(t\).

7 A parachutist of mass \(m \mathrm {~kg}\) opens his parachute when he is moving vertically downwards with a speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after opening his parachute, he has fallen a distance \(x \mathrm {~m}\) from the point where he opened his parachute, and his speed is \(v \mathrm {~ms} ^ { - 1 }\). The forces acting on him are his weight and a resistive force of magnitude \(m v \mathrm {~N}\).

1. Find an expression for \(v\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-12_2715_40_144_2007}
2. Find an expression for \(x\) in terms of \(t\).
3. Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15 \mathrm {~ms} ^ { - 1 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-14_2715_35_143_2012}

7 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(\frac { 200 } { x ^ { 2 } } - \frac { 100 } { x ^ { 3 } }\) for \(x > 0\). When \(t = 0 , x = 1\) and \(P\) has velocity \(10 \mathrm {~ms} ^ { - 1 }\) directed towards \(O\).

1. Show that the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) is given by \(\mathrm { v } = \frac { 10 ( 1 - 2 \mathrm { x } ) } { \mathrm { x } }\).
2. Show that \(x\) and \(t\) are related by the equation \(\mathrm { e } ^ { - 40 \mathrm { t } } = ( 2 \mathrm { x } - 1 ) \mathrm { e } ^ { 2 \mathrm { x } - 2 }\) and deduce what happens to \(x\) as \(t\) becomes large.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A particle \(P\) of mass 0.2 kg is released from rest and falls vertically. At time \(t \mathrm {~s}\) after release \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.8 v \mathrm {~N}\) acts on \(P\).

1. Show that the acceleration of \(P\) is \(( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the value of \(v\) when \(t = 0.6\).

6
\includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-4_238_725_258_712} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.2 kg respectively, are attached to opposite ends of a light inextensible string. \(P\) is placed on a horizontal table and the string passes over a small smooth pulley at the edge of the table. The string is taut and the part of the string attached to \(Q\) is vertical (see diagram). The coefficient of friction between \(P\) and the table is 0.5 . \(Q\) is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at time \(t \mathrm {~s}\) after the instant of projection the speed of the particles is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion of each particle is opposed by a resisting force of magnitude \(0.9 v \mathrm {~N}\). The particle \(P\) does not reach the pulley.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 3 v\).
2. Find the value of \(t\) when the particles have speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance that each particle has travelled in this time.

1 A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of \(P\) is given by \(\mathrm { e } ^ { - 0.5 v } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the velocity of \(P\) when \(t = 1.2\).

3. A particle \(P\) is moving in a straight line. At time \(t\) seconds, \(P\) is at a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 10 } { x + 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) when \(x = 14\) Given that \(x = 2\) when \(t = 1\),
2. find the value of \(t\) when \(x = 14\)

7 A car \(C\) is moving horizontally in a straight line with velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds, where \(v > 0\) and \(t \geqslant 0\). The acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of \(C\) is modelled by the equation
\(a = v \left( \frac { 8 t } { 7 + 4 t ^ { 2 } } - \frac { 1 } { 2 } \right)\).

1. In this question you must show detailed reasoning. Find the times when the acceleration of \(C\) is zero. At \(t = 0\) the velocity of \(C\) is \(17.5 \mathrm {~ms} ^ { - 1 }\) and at \(t = T\) the velocity of \(C\) is \(5 \mathrm {~ms} ^ { - 1 }\).
2. By setting up and solving a differential equation, show that \(T\) satisfies the equation $$T = 2 \ln \left( \frac { 7 + 4 T ^ { 2 } } { 2 } \right)$$
3. Use an iterative formula, based on the equation in part (b), to find the value of \(T\), giving your answer correct to \(\mathbf { 4 }\) significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process.
4. The diagram below shows the velocity-time graph for the motion of \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-06_751_878_1372_322} Find the time taken for \(C\) to decelerate from travelling at its maximum speed until it is travelling at \(5 \mathrm {~ms} ^ { - 1 }\).

5 A particle is moving along a straight line. At time \(t\), the velocity of the particle is \(v\). The acceleration of the particle throughout the motion is \(- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }\), where \(\lambda\) is a positive constant. The velocity of the particle is \(u\) when \(t = 0\). Find \(v\) in terms of \(u , \lambda\) and \(t\).
(7 marks)

7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.

1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
2. Find \(v\) in terms of \(t\).
3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
   [0pt] [2 marks]

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) the body experiences a resistance of magnitude \(( 0.2 + 0.03 v ) \mathrm { N }\) where \(v \mathrm {~ms} ^ { - 1 }\) is its speed.

1. Show that \(v\) satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
2. Find an expression for \(t\) in terms of \(v\).
3. Calculate, to the nearest second, the time taken for the body to come to rest.

5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).

8 A stone, of mass 0.05 kg , is moving along the smooth horizontal floor of a tank, which is filled with oil. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(0.08 v ^ { 2 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.6 v ^ { 2 }$$ (2 marks)
2. The initial speed of the stone is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 15 } { 5 + 24 t }$$ (5 marks)

5 A golf ball, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ground. At time \(t\) seconds, the golf ball has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the golf ball moves, it experiences a resistance force of magnitude \(0.2 m v ^ { \frac { 1 } { 2 } }\) newtons until it comes to rest. No other horizontal force acts on the golf ball. Model the golf ball as a particle.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.2 v ^ { \frac { 1 } { 2 } }$$
2. When \(t = 0\), the speed of the golf ball is \(16 \mathrm {~ms} ^ { - 1 }\). Show that \(v = ( 4 - 0.1 t ) ^ { 2 }\).
3. Find the value of \(t\) when \(v = 1\).
4. Find the distance travelled by the golf ball as its speed decreases from \(16 \mathrm {~ms} ^ { - 1 }\) to \(1 \mathrm {~ms} ^ { - 1 }\).

6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).