Displacement and velocity at given time

A question is this type if and only if it asks to find the displacement and/or velocity of a particle at a specific time t during SHM.

5 questions · Standard +0.5

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Edexcel M3 2008 June Q2
11 marks Standard +0.3
2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(u\).
  2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
  3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).
Edexcel M3 2012 June Q1
9 marks Standard +0.3
  1. A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0 , P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2 \mathrm { e } ^ { - x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing.
    1. Find the acceleration of \(P\) in terms of \(x\).
    2. Find \(x\) in terms of \(t\).
    3. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac { \pi } { 2 }\) seconds. At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\) and \(v = 6\). Find
    4. the greatest distance of \(P\) from \(O\) during the motion,
    5. the greatest magnitude of the acceleration of \(P\) during the motion,
    6. the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\).
OCR M3 2012 January Q6
13 marks Standard +0.8
6 A particle \(P\) starts from rest at a point \(A\) and moves in a straight line with simple harmonic motion. At time \(t \mathrm {~s}\) after the motion starts, \(P\) 's displacement from a point \(O\) on the line is \(x \mathrm {~m}\) towards \(A\). The particle \(P\) returns to \(A\) for the first time when \(t = 0.4 \pi\). The maximum speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\) and occurs when \(P\) passes through \(O\).
  1. Find the distance \(O A\).
  2. Find the value of \(x\) and the velocity of \(P\) when \(t = 1\).
  3. Find the number of occasions in the interval \(0 < t < 1\) at which \(P\) 's speed is the same as that when \(t = 1\), and find the corresponding values of \(x\) and \(t\).
OCR M3 2010 June Q5
11 marks Standard +0.8
5
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-4_234_1003_1007_571} Particles \(P _ { 1 }\) and \(P _ { 2 }\) are each moving with simple harmonic motion along the same straight line. \(P _ { 1 }\) 's motion has centre \(C _ { 1 }\), period \(2 \pi \mathrm {~s}\) and amplitude \(3 \mathrm {~m} ; P _ { 2 }\) 's motion has centre \(C _ { 2 }\), period \(\frac { 4 } { 3 } \pi \mathrm {~s}\) and amplitude 4 m . The points \(C _ { 1 }\) and \(C _ { 2 }\) are 6.5 m apart. The displacements of \(P _ { 1 }\) and \(P _ { 2 }\) from their centres of oscillation at time \(t \mathrm {~s}\) are denoted by \(x _ { 1 } \mathrm {~m}\) and \(x _ { 2 } \mathrm {~m}\) respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x _ { 1 } = 3\) and \(x _ { 2 } = 4\).
  1. State expressions for \(x _ { 1 }\) and \(x _ { 2 }\) in terms of \(t\), which are valid until the particles collide. The particles collide when \(t = 5.99\), correct to 3 significant figures.
  2. Find the distance travelled by \(P _ { 2 }\) before the collision takes place.
  3. Find the velocities of \(P _ { 1 }\) and \(P _ { 2 }\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions.
OCR M3 2015 June Q6
11 marks Standard +0.3
6 A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) towards \(A\). The particle \(P\) is next at rest when \(t = 0.25 \pi\) having travelled a distance of 1.2 m .
  1. Find the maximum velocity of \(P\).
  2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\).
  3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\) 's speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\).