Displacement and velocity at given time

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 \(t = 2 \mathrm {~s}\).

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 \(t\).

\includegraphics{figure_5} 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 \(2\pi\) s and amplitude \(3\) m; \(P_2\)'s motion has centre \(C_2\), period \(\frac{4}{3}\pi\) 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 \(t\) s are denoted by \(x_1\) m and \(x_2\) 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. [3]

The particles collide when \(t = 5.99\), correct to \(3\) significant figures.

1. Find the distance travelled by \(P_2\) before the collision takes place. [4]
2. 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. [4]

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 \(x\) 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 \(P\). [3]
2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\). [4]
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 \(x\). [4]

\(A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after \(0.2\) seconds at point \(B\) whose displacement is \(0.2\) m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x\) m.

1. Sketch a graph of \(x\) against \(t\) for \(0 \leq t \leq 0.4\). [4]
2. Find the displacement of \(P\) from \(M\) at \(0.75\) seconds after release. [2]