Can u average (sine)ing function?

Calculus Level 4

A body is executing 1D motion along the $y$ -axis. It's displacement as a function of time is given by $y=A\sin(wt)$ . Let $v_m$ be maximum speed of the body and $v_\text{avg}$ be average speed of the body in one complete oscillation. If the ratio of $v_m:v_\text{avg}$ is given by $\dfrac{\pi}{a}$ , then find the value of $a^2-7a+10$ .

Details and Assumptions:

Take time period of one complete oscillation, $T=\dfrac{2\pi}{w}$ .

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7 5 -6 None 2 -2

1 solution

Sparsh Sarode
May 19, 2016

$y=Asin(wt)$

$\frac{dy}{dt}=v=Awcos(wt)$

$v_{max}=Aw$

$v_{avg}=\frac{\text{total distance covered}}{\text{total time}}$

total distance covered in 1 oscillation is 4A

$v_{avg}=\frac{4A}{T}=\frac{4Aw}{2π}$

$\frac{v_{max}}{v_{avg}}=\frac{π}{2} \Rightarrow a=2$

$2^2-7(2)+10=0$

hence the ans is $\boxed 0$

No, because $v_{avg}=0$ , what follows from $\frac{1}{T}\int_{0}^{T}\frac{dy}{dt}dt=0$ for the average speed!

Andreas Wendler - 5 years ago

Total displacement is zero.. But not distance

Sparsh Sarode - 5 years ago

Integration of cosine-function is 0. Think that there are analogous motions in positive as in negative direction of the y axis!

I guess your "task" is a bit HUMBUG!!!

Andreas Wendler - 5 years ago

@Andreas Wendler – Speed is always positive even when u take average. I agree that integration of cos function from 0 to $2\pi$ is 0 but I think it should be taken $|cos(wt)|$ since we are only considering the magnitude. Am I right?

Sparsh Sarode - 5 years ago

@Sparsh Sarode – How I know for calculation of the average of a speed one has to consider both positive and negative components during the elapsed time.

Andreas Wendler - 5 years ago

@Andreas Wendler – I don't quite understand your question. I agree with @Sparsh Sarode that one would have to integrate the modulus of the cosine function to get the average speed or do the way that he has done in the solution.

Sudeep Salgia - 5 years ago

@Andreas Wendler – Isn't the question correct? How can u take distance travelled to be negative?

M D - 5 years ago

@Andreas Wendler – There can only be negative values if the question asked for velocity. Speed is the magnitude of velocity and hence cannot be negative.

Tristan Goodman - 2 years, 1 month ago

@Sparsh Sarode – We can also simply integrate as @Andreas Wendler is saying from limits 0 to (pi/4) taking 4 in multiplication outside. Can't we?

RAJ RAJPUT - 5 years ago

@Raj Rajput – You are right. Note that $\int_{0}^{2\pi}cosxdx$ and $4\int_{0}^{\frac{\pi}{4}}cosxdx$ do not yield the same answer. You will get $0$ for the first integral

Sparsh Sarode - 5 years ago

@Sparsh Sarode – Exactly... :)

RAJ RAJPUT - 5 years ago

Same approach here =D
A little hint on Latex: if you want to input text in latex, you can use \text{......}

展豪張 - 5 years ago

Thx.. Didnt see tht..

Sparsh Sarode - 5 years ago

Cool! It keeps the spaces between words now =D

展豪張 - 5 years ago

Can u average (sine)ing function?

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