Motion with exponential force

7 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A single horizontal force of magnitude \(0.16 \mathrm { e } ^ { x } \mathrm {~N}\) acts on \(P\) in the direction \(O P\). The velocity of \(P\) when it is at \(O\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
2. Find the time taken by \(P\) to travel 1.4 m from \(O\).

5 A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\). The only force acting on \(P\) is a resistive force of magnitude \(\left( 4 \mathrm { e } ^ { - x } + 12 \right) \mathrm { e } ^ { - x } \mathrm {~N}\). When \(\mathrm { t } = 0 , \mathrm { x } = 0\) and \(v = 4\).

1. Show by integration that \(\mathrm { v } = \frac { 1 + 3 \mathrm { e } ^ { \mathrm { x } } } { \mathrm { e } ^ { \mathrm { x } } }\).
2. Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-10_510_889_269_580} \(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(\mathrm { ABC } = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.
   1. Find the value of \(e\).
   2. Find the size of angle \(\beta\). \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-12_965_1059_267_502} A uniform cylinder with a rough surface and of radius \(a\) is fixed with its axis horizontal. Two identical uniform rods \(A B\) and \(B C\), each of weight \(W\) and length \(2 a\), are rigidly joined at \(B\) with \(A B\) perpendicular to \(B C\). The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with \(A B\) at an angle \(\theta\) to the horizontal. \(D\) and \(E\) are the midpoints of \(A B\) and \(B C\) respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is \(\mu\). The normal reaction between \(A B\) and the cylinder is \(R\) and the normal reaction between \(B C\) and the cylinder is \(N\).
   3. Find the ratio \(R : N\) in terms of \(\mu\).
   4. Given that \(\mu = \frac { 1 } { 3 }\), find the value of \(\tan \theta\).
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3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).

5 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude \(3 \mathrm { e } ^ { - t } \mathrm {~N}\) directed up a line of greatest slope acts on \(P\), where \(t \mathrm {~s}\) is the time after release.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 7.5 \mathrm { e } ^ { - t } - 5\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) up the plane at time \(t \mathrm {~s}\).
2. Express \(v\) in terms of \(t\).
3. Find the distance of \(P\) from \(O\) when \(v\) has its maximum value.

1 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(t \mathrm { e } ^ { - v } \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after release. Find the velocity of \(P\) when \(t = 2\).

3 A particle \(P\) of mass 6 kg moves in a straight line under the action of a single force of magnitude \(F N\) which acts in the direction of motion of \(P\).
At time \(t\) seconds, where \(t \geqslant 0 , F\) is given by \(\mathrm { F } = \frac { 1 } { 5 - 4 \mathrm { e } ^ { - \mathrm { t } ^ { 2 } } }\).
When \(t = 0\), the speed of \(P\) is \(1.9 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse of the force over the period \(0 \leqslant t \leqslant 2\).
2. Find the speed of \(P\) at the instant when \(t = 2\).
3. Find the work done by the force on \(P\) over the period \(0 \leqslant t \leqslant 2\).

3 A particle of mass 4 kg moves along the \(x\)-axis. At time \(t\) seconds the particle is \(x \mathrm {~m}\) from the origin O and has velocity \(v \mathrm {~ms} ^ { - 1 }\). A driving force of magnitude \(20 t \mathrm { t } ^ { - t } \mathrm {~N}\) is applied in the positive \(x\) direction. Initially \(v = 2\) and the particle is at O .

1. Find, in either order, the impulse of the force over the first 3 seconds and the velocity of the particle after 3 seconds. From time \(t = 3\) a resistive force of magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\) and the driving force are applied until the particle comes to rest.
2. Show that, after the resistive force is applied, the only time at which the resultant force on the particle is zero is when \(t = \ln 40\). Hence find the maximum velocity of the particle during the motion.
3. Given that the time \(T\) seconds at which the particle comes to rest is given by the equation \(T = \sqrt { 121 - 80 \mathrm { e } ^ { - T } ( 1 + T ) }\), without solving the equation deduce that \(T \approx 11\).
4. Use a numerical method to find \(T\) correct to 4 decimal places.

A particle \(P\) of mass \(0.4 \text{ kg}\) is released from rest at a point \(O\) on a smooth plane inclined at \(30°\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \text{ m}\) down the plane, the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8e^{-x} \text{ N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).

1. Show that \(v \frac{dv}{dx} = 5 - 2e^{-x}\). [2]
2. Find \(v\) when \(x = 0.6\). [4]

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

1. Find an expression for \(v\) in terms of \(t\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

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

1. Find an expression for \(v\) in terms of \(t\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass 2.5 kg moves along the positive \(x\)-axis. It moves away from a fixed origin \(O\), under the action of a force directed away from \(O\). When \(OP = x\) metres the magnitude of the force is \(2e^{-0.1x}\) newtons and the speed of \(P\) is \(v\) m s\(^{-1}\). When \(x = 0\), \(v = 2\). Find

1. \(v^2\) in terms of \(x\), [6]
2. the value of \(x\) when \(v = 4\). [3]
3. Give a reason why the speed of \(P\) does not exceed \(\sqrt{20}\) m s\(^{-1}\). [1]