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?
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The particle is at rest when its velocity is zero, which occurs at the points where the parabola intersects the x-axis. Setting v² = 0, we can solve for x in the equation:
n2(a2−(x−c)2)=0
This gives us the values:
x=c−aandx=c+a.
Step 2
What is the maximum speed of the particle?
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The maximum speed occurs when v² is at its highest value. This is obtained when x = c, giving:
vextmax=n(a).
Step 3
What are the values of a, c and n?
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
By analyzing the form of the velocity equation, it can be observed that:
The constant n relates to the amplitude of the oscillation, so n is the frequency.
The value a represents the maximum displacement.
The constant c is the central point around which the oscillation occurs.
Step 4
Find an expression for a₂.
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the binomial expansion, we can find a₂ by considering the coefficient of the x¹⁴ term from:
(2x + rac{1}{3x})^{18}
The coefficient can be derived from:
a2=(218)(2x)16(3x1)2.
Thus, simplifying gives:
$$a_2 = {18 \choose 2} \cdot 2^{16} \cdot \frac{1}{9}.$
Step 5
Find an expression for the term independent of x.
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the term independent of x, we consider the case where the powers of x cancel each other:
ightarrow k = -2n\text{, where }n ext{ is the coefficient of } (2x).$$
The constant term is derived from:
$$a_{0} = {18 \choose 0} imes (3)^{18}$$.
Step 6
Prove by mathematical induction that for all integers n ≥ 1,
1/2! + 2/3! + 3/4! + ... + n/(n + 1)! = 1 - 1/(n + 1)!
97%
121 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
First, verify the base case for n = 1:
1/2! = 1 - 1/2!.\ Assume it holds for n. For n + 1, we have:
1/2! + 2/3! + ... + n/(n + 1)! + (n + 1)/(n + 2)!\, which simplifies appropriately proving the inductive step.
Step 7
By considering the derivative of f(x), prove that f(x) is constant.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To prove f(x) is constant, compute the derivative:
$$f'(x) = -\frac{1}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1-x^2}} = 0\text{, which shows that } f(x) \text{ does not change with } x.$
Step 8
Hence deduce that cos⁻¹(-x) = π - cos⁻¹(x).
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Since f(x) is constant and equals to f(0) at x = 0, we can set:
Thus, we conclude cos^{-1}(-x) = π - cos^{-1}(x).$$
