Time to travel between positions

6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).

1. Find the distance \(A B\).
   (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)

The length of \(A B\) is 0.6 m .

1. Find the maximum speed of \(P\)
2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)

7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\). The other end of the spring is attached to a fixed point \(A\). The particle is hanging freely in equilibrium at the point \(B\), where \(A B = 1.5 l\)

1. Show that the modulus of elasticity of the spring is \(2 m g\). The particle is pulled vertically downwards from \(B\) to the point \(C\), where \(A C = 1.8 \mathrm { l }\), and released from rest.
2. Show that \(P\) moves in simple harmonic motion with centre \(B\).
3. Find the greatest magnitude of the acceleration of \(P\). The midpoint of \(B C\) is \(D\). The point \(E\) lies vertically below \(A\) and \(A E = 1.2 l\)
4. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

3 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 6 \mathrm {~m}\) and \(O B = 8 \mathrm {~m}\), with \(O\) between \(A\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). When \(P\) is at \(A\) its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when \(P\) is at \(B\) its speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is 10 m and find the period of the motion. Find the time taken by \(P\) to travel directly from \(A\) to \(B\), through \(O\).

2 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.5 \mathrm {~m}\) and \(O B = 0.75 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) of mass \(m\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The ratio of the kinetic energy of \(P\) when it is at \(A\) to its kinetic energy when it is at \(B\) is \(12 : 11\). Find the amplitude of the motion. Given that the greatest speed of \(P\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 A particle \(P\) moves on a straight line \(A O B\) in simple harmonic motion. The centre of the motion is \(O\), and \(P\) is instantaneously at rest at \(A\) and \(B\). The point \(C\) is on the line \(A O B\), between \(A\) and \(O\), and \(C O = 10 \mathrm {~m}\). When \(P\) is at \(C\), the magnitude of its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and it is moving towards \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the period of the motion, in terms of \(\pi\),
2. the amplitude of the motion. The point \(M\) is the mid-point of \(O B\). Find the time that \(P\) takes to travel directly from \(C\) to \(M\).

2 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 \(3 : 4\).

1. Find the amplitude of the motion.
   The maximum speed of \(P\) during its motion is \(\frac { 1 } { 3 } \pi \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the period of the motion.
3. Find the time taken for \(P\) to travel directly from \(A\) to \(B\).

3 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 \(A\).

1. Find \(d\).
   The period of the motion is \(\pi \mathrm { s }\) and the maximum acceleration of \(P\) during the motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of \(P\) when it is 7 m from \(O\).
3. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 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 \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of \(P\) when it is at \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-04_64_1566_492_328}
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 The point \(O\) is on the fixed line \(l\). The point \(A\) on \(l\) is such that \(O A = 3 \mathrm {~m}\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\) and period \(\pi\) seconds. When \(P\) is at \(A\) its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(P\) when it is at the point \(B\) on \(l\), where \(O B = 6 \mathrm {~m}\) and \(B\) is on the same side of \(O\) as \(A\). Find, correct to 2 decimal places, the time, in seconds, taken for \(P\) to travel directly from \(A\) to \(B\).

1 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

1 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

1 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

3. A particle is performing simple harmonic motion along a straight line between the points \(A\) and \(B\) where \(A B = 8 \mathrm {~m}\). The period of the motion is 12 seconds.

1. Find the maximum speed of the particle in terms of \(\pi\). The points \(P\) and \(Q\) are on the line \(A B\) at distances of 3 m and 6 m respectively from \(A\).
2. Find, correct to 3 significant figures, the time it takes for the particle to travel directly from \(P\) to \(Q\).
   (6 marks)

4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).

1. For the motion of the particle write down
   1. the period,
   2. the amplitude.
2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
   (4 marks)

8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).

1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).

\includegraphics{figure_5} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac{1}{2}mg\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m\), \(g\) and \(\beta\), and hence show that \(\beta = \frac{1}{4}\pi\). [4] The particle is given a velocity of magnitude \(\sqrt{(ag)}\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot{x} = -\frac{2g}{a}x.$$ [3] Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time. [6]

The points \(O\), \(A\), \(B\) and \(C\) lie in a straight line, in that order, where \(OA = 0.6\) m, \(OB = 0.8\) m and \(OC = 1.2\) m. A particle \(P\), moving along this straight line, has a speed of \(\left(\frac{1}{10}\sqrt{5}\right)\) m s\(^{-1}\) at \(A\), \(\left(\frac{1}{5}\sqrt{5}\right)\) m s\(^{-1}\) at \(B\) and is instantaneously at rest at \(C\).

1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). [5]

Given that \(P\) is performing simple harmonic motion with centre \(O\),

1. show that the speed of \(P\) at \(O\) is 0.6 m s\(^{-1}\), [2]
2. find the magnitude of the acceleration of \(P\) as it passes \(A\), [2]
3. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). [4]

In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(AB = 4L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(PQ\), where \(PQ = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).

1. Show that \(b = (2 - \sqrt{2})L\). [5]
2. Find the speed of \(C\) as it passes \(P\). [2]

\includegraphics{figure_5} For advanced players, the window \(PQ\) is moved to the centre of \(AB\) so that \(AP = QB\), as shown in Figure 5.

1. Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position. [3]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5 \sin (\frac{1}{4}\pi t)\).

1. Prove that \(P\) is moving with simple harmonic motion. [3]
2. Find the period and the amplitude of the motion. [2]
3. Find the maximum speed of \(P\). [2]

The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(OA = 2\) m and \(OB = 3\) m.

1. Find the time taken by \(P\) to travel directly from \(A\) to \(B\). [4]

A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\) with period 2 s. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(v = 0\) and \(P\) is at a point \(A\) where \(OA = 0.25\) m. Find the smallest positive value of \(t\) for which \(AP = 0.375\) m. [6]

\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.

1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]

Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).

1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]

11 Answer only one of the following two alternatives.
EITHER
A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic string of natural length \(l\), and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is \(\frac { 13 } { 7 } l\). The particle is released from rest in this position and reaches its greatest height when the length of the string is \(\frac { 11 } { 7 } l\).

1. Show that the modulus of elasticity of the string is \(\frac { 7 } { 5 } \mathrm { mg }\).
2. Show that \(P\) moves in simple harmonic motion about the equilibrium position and state the period of the motion.
3. Find the time after release when the speed of \(P\) is first equal to half of its maximum value.
   OR
   For a random sample of 12 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) and the equation of the regression line of \(x\) on \(y\) are $$y = b x + 4.5 \quad \text { and } \quad x = a y + c$$ respectively, where \(a , b\) and \(c\) are constants. The product moment correlation coefficient for the sample is 0.6 .
4. Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
5. Given that \(b - a = 0.5\), find the values of \(a\) and \(b\).
6. Given that the sum of the \(x\)-values in the sample data is 66, find the value of \(c\) and sketch the two regression lines on the same diagram. For each of the 12 pairs of values of \(( x , y )\) in the sample, another variable \(z\) is considered, where \(z = 5 y\).
7. State the coefficient of \(x\) in the equation of the regression line of \(z\) on \(x\) and find the value of the product moment correlation coefficient between \(x\) and \(z\), justifying your answer.