Photo AI
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?
Answer
To determine when the particle is at rest, we look for the value of x where the velocity v is equal to zero. From the given graph of v² as a function of x, we find the value of x where the parabola intersects the x-axis. In this case, the intersections indicate v² = 0, which corresponds to v = 0. From the graph, it appears that the particle is at rest at x = 3 and x = 7.
Step 2
What is the maximum speed of the particle?
Answer
The maximum speed occurs at the vertex of the parabola representing v². By analyzing the graph, we can find the maximum value of v². The maximum speed v can be calculated as the square root of the maximum value of v², represented by the height of the parabola above the x-axis. In this case, the maximum speed is when x is at its midpoint (x = 5). Plugging this into the equation gives a maximum speed of v = 1 m/s.
Step 3
What are the values of a, c and n?
Answer
In the equation v² = n²(a² - (x - c)²), we compare it with the standard form of a parabola. From the context provided, we can see that a represents the amplitude of the oscillation, n stands for the frequency, and c is the phase shift. Comparing coefficients allows us to determine these constants. Therefore:
- a = 2
- c = 5
- n = 1.
Step 4
Step 5
Find an expression for the term independent of x.
Answer
To find the term that is independent of x in the expansion of (2x + 1/3x)¹⁸, we need to set the powers of x to zero. This occurs when the exponents of x cancel out. Thus, we can substitute: a₀ = inom{18}{9} imes (2)^9 imes (1/3)^9 = 48620 imes rac{512}{19683}.
Step 6
Prove by mathematical induction that for all integers n ≥ 1, ...
Answer
We start with the base case of n = 1. Substituting this into the equation verifies the statement holds. For the inductive step, assume it holds for n = k, and then demonstrate it holds for n = k + 1. This involves some algebraic manipulation which leads us to show:
a_{k + 1} = a_k + f(k), leading to completion for all integers n ≥ 1.
Step 7
By considering the derivative of f(x), prove that f(x) is constant.
Answer
To prove that f(x) = cos⁻¹(x) + cos⁻¹(-x) is constant, we differentiate f(x) with respect to x:
f'(x) = - rac{1}{ m { ext{ ext{sqrt}}(1-x^2)}} + rac{1}{ m { ext{ ext{sqrt}}(1-(-x)^2)}}, which simplifies to show that the derivative is always zero. Hence, f(x) does not change and is constant.
Step 8