Variable force (position x) - find velocity

7 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves towards a fixed point \(A\) on the line. At time \(t \mathrm {~s}\) the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude (5-x) N acts on \(P\) in the direction towards \(O\).

1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2 x\).
2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\).

5 A particle \(P\) of mass 0.4 kg is released from rest at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The motion of \(P\) down the slope is opposed by a force of magnitude \(0.6 x \mathrm {~N}\), where \(x \mathrm {~m}\) is the distance \(P\) has travelled down the slope. \(P\) comes to rest before reaching the foot of the slope. Calculate

1. the greatest speed of \(P\) during its motion,
2. the distance travelled by \(P\) during its motion.

1 A particle \(P\) of mass 0.6 kg is projected horizontally with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(0.3 x \mathrm {~N}\) acts on \(P\) in the direction \(O P\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\). Calculate the velocity of \(P\) when \(x = 8\).

3 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\), and moves in a straight line on a smooth horizontal surface. A force of magnitude \(0.4 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Given that \(P\) comes to instantaneous rest when \(x = 2.5\), find the initial kinetic energy of \(P\).
2. Find the value of \(x\) on the first occasion when the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement is \(x \mathrm {~m}\) from \(O\). The force acting on \(B\) has magnitude \(\left( 2 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and is directed horizontally away from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 8\).
2. Find the velocity of \(B\) when \(x = 1.5\). An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 6 - 3 x$$
3. Find the magnitude of this extra force and state the direction in which it acts.

6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.

3. A particle \(P\) of mass \(m \mathrm {~kg}\) slides from rest down a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved a distance \(x\) metres down the plane, the resistance to the motion of \(P\) from non-gravitational forces has magnitude \(m x ^ { 2 }\) newtons. Find

1. the speed of \(P\) when \(x = 2\),
2. the distance \(P\) has moved when it comes to rest for the first time.

3 At time \(t = 0 \mathrm {~s}\) a particle \(P\), of mass 0.3 kg , is 1 m away from a point \(O\) on a smooth horizontal plane and is moving away from \(O\) with speed \(\sqrt { 5 } \mathrm {~ms} ^ { - 1 }\). The only horizontal force acting on \(P\) has magnitude \(1.5 x \mathrm {~N}\), where \(x\) is the distance \(O P\), and acts away from \(O\).

1. Show that the speed of \(P , v \mathrm {~ms} ^ { - 1 }\), is given by \(v = \sqrt { 5 } x\).
2. Find an expression for \(v\) in terms of \(t\).

2. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force directed away from the origin \(O\). When \(P\) is \(x\) metres from \(O\), the magnitude of the force is \(3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and \(P\) has a speed of \(v \mathrm {~ms} ^ { - 1 }\). Given that when \(x = 1 , P\) is moving away from \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),

1. find an expression for \(v ^ { 2 }\) in terms of \(x\),
2. show that when \(x = 4 , P\) has a speed of \(7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 1 decimal place.

A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \text{ m s}^{-1}\) and moves towards a fixed point \(A\) on the line. At time \(t\) s the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v \text{ m s}^{-1}\). A resistive force of magnitude \((5 - x)\) N acts on \(P\) in the direction towards \(O\).

1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2x\). [6]
2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\). [5]

A particle \(P\) of mass \(0.1\) kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude \(0.2e^{-x}\) N acts on \(P\), where \(x\) m is the displacement of \(P\) from a fixed point \(O\) on the line. The velocity of \(P\) is \(v\) m s\(^{-1}\) when its displacement from \(O\) is \(x\) m.

1. Show that $$v\frac{dv}{dx} = ke^{-x},$$ where \(k\) is a constant to be found. [2]

\(P\) passes through \(O\) with velocity \(2.2\) m s\(^{-1}\).

1. Calculate the value of \(x\) at the instant when the velocity of \(P\) is \(2\) m s\(^{-1}\). [4]
2. Show that the speed of \(P\) does not fall below \(0.917\) m s\(^{-1}\), correct to \(3\) significant figures. [2]

\(O\) and \(A\) are fixed points on a rough horizontal surface, with \(OA = 1 \text{ m}\). A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally with speed \(U \text{ m s}^{-1}\) from \(A\) in the direction \(OA\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \text{ m}\), the velocity of \(P\) is \(v \text{ m s}^{-1}\). The coefficient of friction between the surface and \(P\) is \(0.4\). A force of magnitude \(\frac{0.8}{x} \text{ N}\) acts on \(P\) in the direction \(PO\).

1. Show that, while the particle is in motion, \(v \frac{\text{d}v}{\text{d}x} = -4 - \frac{2}{x}\). [3]

It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).

1. Find the set of possible values of \(U\). [5]

A particle \(P\) of mass \(0.3\text{ kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\text{ m s}^{-1}\). A force of magnitude \(2x\text{ N}\) acts on \(P\) in the direction \(PO\), where \(x\text{ m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = kx\) and state the value of the constant \(k\). [2]
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]

A particle \(P\) of mass \(0.3\,\text{kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\,\text{m s}^{-1}\). A force of magnitude \(2x\,\text{N}\) acts on \(P\) in the direction \(PO\), where \(x\,\text{m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v\frac{dv}{dx} = kx\) and state the value of the constant \(k\). [2]
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]

A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v\) m s\(^{-1}\) when its displacement is \(x\) m from \(O\). The force acting on \(B\) has magnitude \((2 + 0.3x^2)\) N and is directed horizontally away from \(O\).

1. Show that \(v\frac{dv}{dx} = 1.2x^2 + 8\). [2]
2. Find the velocity of \(B\) when \(x = 1.5\). [3]

An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v\frac{dv}{dx} = 1.2x^2 + 6 - 3x.$$

1. Find the magnitude of this extra force and state the direction in which it acts. [2]

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

1. Show that the velocity \(v\) m s\(^{-1}\) of \(P\) is given by \(v = \frac{10(1-2x)}{x}\). [5]
2. Show that \(x\) and \(t\) are related by the equation \(e^{-40t} = (2x-1)e^{2x-2}\) and deduce what happens to \(x\) as \(t\) becomes large. [5]

A particle \(P\) of mass \(2\) kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and the displacement of \(P\) from \(O\) is \(x\) m. A force of magnitude \(\left(8x - \frac{128}{x^3}\right)\) N acts on \(P\) in the direction \(OP\). When \(t = 0\), \(x = 8\) and \(v = -15\).

1. Show that \(v = -\frac{2}{3}(x^2 - 4)\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(0.5\) kg moves in a straight line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The only forces acting on \(P\) are a force of magnitude \(\frac{150}{(x+1)^2}\) N in the direction of increasing displacement and a resistive force of magnitude \(\frac{450}{(x+1)^3}\) N. When \(t = 0\), \(x = 0\) and \(v = 20\). Find \(v\) in terms of \(x\), giving your answer in the form \(v = \frac{Ax + B}{(x + 1)}\), where \(A\) and \(B\) are constants to be determined. [6]

A toy car of mass \(0.2\) kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time \(t = 0\) the car passes through a fixed point \(O\). After \(t\) seconds the speed of the car is \(v \text{ m s}^{-1}\) and the car is at a point \(P\) with \(OP = x\) metres. The resultant force on the car is modelled as \(\frac{1}{5}x(4 - 3x)\) N in the direction \(OP\). The car comes to instantaneous rest when \(x = 6\). Find

1. an expression for \(v^2\) in terms of \(x\), [7]
2. the initial speed of the car. [2]