The point \(P\) is 4 metres to the right of the origin \(O\) on a straight line.
A particle is released from rest at \(P\) and moves along the straight line in simple harmonic motion about \(O\), with period \(8 \pi\) seconds.
After \(2 \pi\) seconds, another particle is released from rest at \(P\) and also moves along this straight line in simple harmonic motion about \(O\), with period \(8 \pi\) seconds.
Find when and where the two particles first collide. (3 marks)
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\(\text{Particle is at rest at point}\ P\ \text{and moving in SHM} \)
\(\Rightarrow P\ \text{is at extremity of motion} \)
\(\text{Period}\ = 8\pi = \dfrac{2\pi}{n}\ \Rightarrow \ n=\dfrac{1}{4} \)
\(\text{Amplitude}\ =4 \)
\(\text{Particle 1:}\)
\(x_1=a\,\cos(nt)=4\,\cos\,\dfrac{t}{4} \)
\(\text{Particle 2:}\)
| \(x_2\) | \(=4\,\cos\,\Big(\dfrac{t-2\pi}{4}\Big) \) | |
| \(=4\,\cos\,\Big(\dfrac{2\pi-t}{4}\Big) \) | ||
| \(=4\,\cos\,\Big(\dfrac{\pi}{2}-\dfrac{t}{4}\Big) \) | ||
| \(=4\,\sin\,\Big(\dfrac{t}{4}\Big) \) |
\(\text{Collision when}\ \ x_1=x_2: \)
| \(4\,\sin\,\Big(\dfrac{t}{4}\Big) \) | \(=4\,\cos\,\Big(\dfrac{t}{4}\Big) \) | |
| \(\tan\,\Big(\dfrac{t}{4}\Big) \) | \(=1\) | |
| \(\dfrac{t}{4}\) | \(=\dfrac{\pi}{4}, \dfrac{5\pi}{4} \) | |
| \(t\) | \(=\pi, 5\pi, … \) |
\(\text{1st collision occurs at}\ \ t=5\pi \ \text{seconds}\ \ (t\geq 2\pi)\)
\(\text{Position of collision}\)
\(\text{Find}\ x_1\ \text{when}\ t=5\pi : \)
\(x_1=4\,\cos\,\dfrac{5\pi}{4}=-\dfrac{4}{\sqrt2} = -2\sqrt2 \ \text{m}\ \ (\text{or}\ 2\sqrt2\ \text{m to left of origin}) \)