A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(A B=6 \mathrm{~m}\). The period of the motion is \(\frac{1}{2} \pi \mathrm{~s}\). Find the speed of \(P\) when it is 2 m from \(B\).

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(O A=3.5 \mathrm{~m}\) and \(O B=1 \mathrm{~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm{~m} \mathrm{~s}^{-2}\).
(i) Find the speed of \(P\) when it is at \(O\).

(ii) Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A=0.1 \mathrm{~m}\) and \(O B=0.5 \mathrm{~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion.

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is 2.5 m . The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(O L=1.5 \mathrm{~m}\). The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio 3:4.
(i) Find the distance \(O M\).

The time taken by \(P\) to travel directly from \(L\) to \(M\) is 2 s .
(ii) Find the period of the motion.

(iii) Find the speed of \(P\) when it passes through \(L\).

The piston in a large engine rises and falls in simple harmonic motion. When the piston is 1.6 m below its highest level, the rate of change of its height is \(\frac{3}{5} \pi\) metres per second. When the piston is 0.2 m below its highest level, the rate of change of its height is \(\frac{1}{4} \pi\) metres per second. Find the amplitude and period of the motion.

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line, on opposite sides of \(O\), with \(O A=1.6 \mathrm{~m}\) and \(O B=1.2 \mathrm{~m}\). The ratio of the speed of \(P\) at \(A\) to its speed at \(B\) is \(3: 4\).
(i) Find the amplitude of the motion.

The maximum speed of \(P\) during its motion is \(\frac{1}{3} \pi \mathrm{~m} \mathrm{~s}^{-1}\).
(ii) Find the period of the motion.

(iii) Find the time taken for \(P\) to travel directly from \(A\) to \(B\).

A particle \(P\) moves on the positive \(x\)-axis in simple harmonic motion. The centre of the motion is a distance \(d \mathrm{~m}\) from the origin \(O\), where \(0<d<6.5\). The points \(A\) and \(B\) are on the positive \(x\)-axis, with \(O A=6.5 \mathrm{~m}\) and \(O B=7.5 \mathrm{~m}\). The magnitude of the acceleration of \(P\) when it is at \(B\) is twice the magnitude of the acceleration of \(P\) when it is at \(A\).
(i) Find \(d\).

The period of the motion is \(\pi \mathrm{s}\) and the maximum acceleration of \(P\) during the motion is \(10 \mathrm{~m} \mathrm{~s}^{-2}\).
(ii) Find the speed of \(P\) when it is 7 m from \(O\).

(iii) Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

A particle \(P\) of mass \(m \mathrm{~kg}\) is projected vertically upwards from a point \(O\) with an initial speed of \(20 \mathrm{~ms}^{-1}\) and moves under gravity. There is a resistive force of magnitude \(0.025 m v^{2} \mathrm{~N}\), where \(v \mathrm{~ms}^{-1}\) is the speed of \(P\) at time \(t \mathrm{~s}\) after projection. The displacement of \(P\) from \(O\) is \(x \mathrm{~m}\) at time \(t \mathrm{~s}\) after projection. (a) Find an expression for \(v\) in terms of \(x\), while \(P\) is moving upwards. (b) Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards.

A particle \(P\) of mass \(8\,\text{kg}\) moves in a straight horizontal line. At time \(t\) s it has displacement \(x\) m from \(O\) and velocity \(v\,\text{m s}^{-1}\). The only horizontal force has magnitude \(x^3+4x\) N and acts in the direction \(OP\). Initially, \(t=0\), \(x=0\), and \(v=1\).

(a) Find \(v\) in terms of \(x\), in the form \(v=ax^2+b\).

(b) Find \(x\) in terms of \(t\).

A ball of mass \(m\) kg is projected vertically upwards with initial speed \(U\,\text{m s}^{-1}\). At time \(t\), it has travelled distance \(x\) m and has speed \(v\,\text{m s}^{-1}\). There is a resistive force of magnitude \(mkv^2\) N, where \(k\gt0\).

(a) Show that, while the ball is moving upwards,

(b) Given \(k=0.025\) and \(U=20\), find the time taken to reach maximum height.

A particle \(P\) of mass \(m\,\text{kg}\) is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\,\text{m s}^{-1}\) is the velocity of \(P\) at time \(t\) s.

(a) Find an expression for \(v\) in terms of \(t\).

The displacement of \(P\) from \(O\) at time \(t\) is \(x\) m.

(b) Find an expression for \(v^2\) in terms of \(x\).

A particle \(P\) of mass \(0.5\,\text{kg}\) moves in a straight line. At time \(t\) seconds, the velocity of \(P\) is \(v\,\text{m s}^{-1}\), and its displacement from a fixed point \(O\) on the line is \(x\) metres.

The only forces acting on \(P\) are a force of magnitude \(\dfrac{150}{(x+1)^2}\,\text{N}\) in the direction of increasing displacement and a resistive force of magnitude \(\dfrac{450}{(x+1)^3}\,\text{N}\).

When \(t=0\), \(x=0\) and \(v=20\).

Find \(v\) in terms of \(x\), giving your answer in the form \(v=\dfrac{Ax+B}{x+1}\), where \(A\) and \(B\) are constants to be determined.

A particle \(P\) moving in a straight line has displacement \(x \mathrm{~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\) 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 \(v=72\).
(a) Find an expression for \(v\) in terms of \(x\).
(b) Find an expression for \(x\) in terms of \(t\).

A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t \mathrm{~s}\) the velocity of \(P\) is \(v \mathrm{~m} \mathrm{~s}^{-1}\) 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 \(t=0, x=0\) and \(v=4\).
(a) Show by integration that \(v=\frac{1+3 \mathrm{e}^{x}}{\mathrm{e}^{x}}\).

(b) Find an expression for \(x\) in terms of \(t\).

A particle \(P\) of mass 5 kg moves along a horizontal straight line. At time \(t \mathrm{~s}\), the velocity of \(P\) is \(v \mathrm{~m} \mathrm{~s}^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x \mathrm{~m}\). The forces acting on \(P\) are a force of magnitude \(\frac{500}{v} \mathrm{~N}\) in the direction \(O P\) and a resistive force of magnitude \(\frac{1}{2} v^{2} \mathrm{~N}\). When \(t=0, x=0\) and \(v=5\).
(a) Find an expression for \(v\) in terms of \(x\).
(b) State the value that the speed approaches for large values of \(x\).

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 \(t \mathrm{~s}\) the velocity of \(P\) is \(v \mathrm{~ms}^{-1}\) and the displacement of \(P\) from \(O\) is \(x \mathrm{~m}\).
A force of magnitude \(\left(8 x-\frac{128}{x^{3}}\right) \mathrm{N}\) acts on \(P\) in the direction \(O P\). When \(t=0, x=8\) and \(v=-15\).
(a) Show that \(v=-\frac{2}{x}\left(x^{2}-4\right)\).
(b) Find an expression for \(x\) in terms of \(t\).

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 \(x\gt 0\). When \(t=0, x=1\) and \(P\) has velocity \(10 \mathrm{~ms}^{-1}\) directed towards \(O\).
(a) Show that the velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\) of \(P\) is given by \(v=\frac{10(1-2 x)}{x}\).
(b) Show that \(x\) and \(t\) are related by the equation \(\mathrm{e}^{-40 t}=(2 x-1) \mathrm{e}^{2 x-2}\) and deduce what happens to \(x\) as \(t\) becomes large.

A particle \(P\) of mass \(m \mathrm{~kg}\) moves in a horizontal straight line against a resistive force of magnitude \(m k v^{2} \mathrm{~N}\), where \(v \mathrm{~ms}^{-1}\) is the speed of \(P\) after it has moved a distance \(x \mathrm{~m}\) and \(k\) is a positive constant. The initial speed of \(P\) is \(u \mathrm{~ms}^{-1}\).
(a) Show that \(x=\frac{1}{k} \ln 2\) when \(v=\frac{1}{2} u\).

Beginning at the instant when the speed of \(P\) is \(\frac{1}{2} u\), an additional force acts on \(P\). This force has magnitude \(\frac{5 m}{v} \mathrm{~N}\) and acts in the direction of increasing \(x\).
(b) Show that when the speed of \(P\) has increased again to \(u \mathrm{~ms}^{-1}\), the total distance travelled by \(P\) is given by an expression of the form
\(\frac{1}{3 k} \ln \left(\frac{A-k u^{3}}{B-k u^{3}}\right),\)
stating the values of the constants \(A\) and \(B\).