A particle is moving along the -x-axis in simple harmonic motion. The displacement of the particle is x metres and its velocity is v m s⁻¹. The parabola below shows v² as a function of x. (i) For what value(s) of x is the particle at rest? (ii) What is the maximum speed of the particle? (iii) The velocity v of the particle is given by the equation v² = n²(a² - (x - c)²) where a, c and n are positive constants. What are the values of a, c and n?

SSCE HSC Mathematics Extension 1 Mathematical induction involving series: A particle is moving along the -

A particle is moving along the -x-axis in simple harmonic motion - HSC - SSCE Mathematics Extension 1 - Question 13 - 2015 - Paper 1

Question 13

A particle is moving along the -x-axis in simple harmonic motion. The displacement of the particle is x metres and its velocity is v m s⁻¹. The parabola below shows ... show full transcript

Worked Solution & Example Answer:A particle is moving along the -x-axis in simple harmonic motion - HSC - SSCE Mathematics Extension 1 - Question 13 - 2015 - Paper 1

Step 1

For what value(s) of x is the particle at rest?

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Answer

The particle is at rest when its velocity v is zero. From the equation v² = n²(a² - (x - c)²), setting v² to zero gives:

$0 = n²(a² - (x - c)²)$

Solving for x results in:

$(x - c)² = a²$

Taking the square root of both sides gives:

$x - c = ±a$

Thus, the values of x where the particle is at rest are:

$x = c + a$ and $x = c - a$ .

Step 2

What is the maximum speed of the particle?

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Answer

The maximum speed occurs when the displacement x is zero, located at the maximum value of v² on the graph. From the equation for v², substituting x = c results in:

$v² = n²(a² - (c - c)²) = n²a²$

Thus, the maximum speed v is given by:

$v = n a$ .

Step 3

What are the values of a, c and n?

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Answer

The equation provided indicates the form of simple harmonic motion, where:

a is the amplitude of the motion,
c is the equilibrium position (the x-coordinate of the vertex of the parabola),
n relates to the angular frequency of the motion. Thus:
a corresponds to half the width of the parabola at the maximum height,
c is the x-value at the vertex of the parabola,
n is derived from the coefficient that relates to the speed at maximum displacement.

A particle is moving along the -x-axis in simple harmonic motion - HSC - SSCE Mathematics Extension 1 - Question 13 - 2015 - Paper 1

Worked Solution & Example Answer:A particle is moving along the -x-axis in simple harmonic motion - HSC - SSCE Mathematics Extension 1 - Question 13 - 2015 - Paper 1

For what value(s) of x is the particle at rest?

What is the maximum speed of the particle?

What are the values of a, c and n?

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