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#1 2021-05-02 06:55:50

mathland
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Registered: 2021-03-25
Posts: 438

Speed of An Object

The distance s (in meters) of an
object from the origin at time t (in seconds) is modeled by the
function s(t) = (1/8) cost.

When is the speed of the object at a maximum?

Tell me the steps.

Thanks.

Last edited by mathland (2021-05-02 06:57:48)

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#2 2021-05-02 07:38:37

zetafunc
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Re: Speed of An Object

You have been given the displacement, s, of the object in terms of time, t. If you differentiate this with respect to t, you'll get the object's velocity in terms of time. How would you find the maximum speed from here?

Note that this equation describes a phenomenon known as simple harmonic motion -- if you've come across this before (and know the sort of motion it's describing), you can have a think about how the object's speed varies between its points of maximum and minimum displacement, which is a shortcut to the answer.

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#3 2021-05-03 09:17:40

mathland
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Re: Speed of An Object

zetafunc wrote:

You have been given the displacement, s, of the object in terms of time, t. If you differentiate this with respect to t, you'll get the object's velocity in terms of time. How would you find the maximum speed from here?

Note that this equation describes a phenomenon known as simple harmonic motion -- if you've come across this before (and know the sort of motion it's describing), you can have a think about how the object's speed varies between its points of maximum and minimum displacement, which is a shortcut to the answer.


Let me see.

Let V = velocity

We can then say that V = s'(t).

Taking the derivative in terms of t, I get

V = -(1/8) sin(t).

I don't know how to find the maximum speed from this point on.
I am not too familiar with simple harmonic motion except that it is related
to waves, like ocean waves. Yes?

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#4 2021-05-03 11:05:41

zetafunc
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Registered: 2014-05-21
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Re: Speed of An Object

mathland wrote:

Taking the derivative in terms of t, I get

V = -(1/8) sin(t).

I don't know how to find the maximum speed from this point on.

What is the range of possible values of sin(t)?

What is the range of possible values of -(1/8)sin(t)?

What is the range of possible values of V?

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#5 2021-05-03 11:29:05

mathland
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Registered: 2021-03-25
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Re: Speed of An Object

zetafunc wrote:
mathland wrote:

Taking the derivative in terms of t, I get

V = -(1/8) sin(t).

I don't know how to find the maximum speed from this point on.

What is the range of possible values of sin(t)?

What is the range of possible values of -(1/8)sin(t)?

What is the range of possible values of V?

Honestly, I never quite understood the range idea. I think you mean the y-value. I see the point (x, y) in terms of (domain, range).

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#6 2021-05-03 19:26:15

Bob
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Registered: 2010-06-20
Posts: 9,128

Re: Speed of An Object

You're right that domain is used for the x values and range for the ys.

But range is also used, as here, as a general term for any variable., x or y.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#7 2021-05-04 01:46:32

mathland
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Re: Speed of An Object

Bob wrote:

You're right that domain is used for the x values and range for the ys.

But range is also used, as here, as a general term for any variable., x or y.

Bob

The textbook answer is (pi/2) + pi•n, where n is an integer.

Why is that the answer?

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#8 2021-05-04 05:44:12

Bob
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Registered: 2010-06-20
Posts: 9,128

Re: Speed of An Object

Firstly, note the use of the word speed, not velocity.  Velocity is a vector variable meaning that direction is important as well as magnitude.  Speed is a scalar, meaning that only magnitude matters.

So the maximum speed occurs when the ds/dt term has a maximum or a minimum.  It might seem odd that the minimum also counts, but on the graph a minimum means a highest but negative value.

So we are looking for the values of t that make |(-1/8)sin(t)| maximum.  The || lines here mean the 'absolute value of' ignoring the sign of the answer.

We cannot do much about -1/8 as its fixed so the only chance to vary the speed comes from the sin(t) term.

Look at https://www.mathsisfun.com/algebra/trig … raphs.html

This shows the graph of a sine function.  It has a maximum positive value at t = pi/2 (measured in radians**) and a minimum at t = 3pi/2.  And it is a periodic function, meaning the values repeat every 2pi.

So there are many answers; every time a max or min occurs.  We could write the answer like this:

pi/2;  3pi/2;  5pi/2;  7pi/2 etc etc.

A neater way to say this is to give the first value, pi/2, and then give a term that describes all the repeats in one go.  The repeats are pi apart so n.pi does this (where n is any integer).  Writing the answer in this way is called the general answer taking full account or the periodic nature of the sine curve.

** I'm hoping you know about radians.  If not read on:

When you first meet angles you are taught that there are 360 degrees in a full turn.  This way of writing angles is not the only way.  In calculus d(cosx)dx = -sinx only if the angles are measured in radians. The conversion is:

pi radians = 180 degrees.

so sin(90°) = sin(pi/2) which is why the answer is given with t in  radians not degrees.

More here: https://www.mathsisfun.com/geometry/radians.html

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#9 2021-05-04 07:47:17

mathland
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Registered: 2021-03-25
Posts: 438

Re: Speed of An Object

Bob wrote:

Firstly, note the use of the word speed, not velocity.  Velocity is a vector variable meaning that direction is important as well as magnitude.  Speed is a scalar, meaning that only magnitude matters.

So the maximum speed occurs when the ds/dt term has a maximum or a minimum.  It might seem odd that the minimum also counts, but on the graph a minimum means a highest but negative value.

So we are looking for the values of t that make |(-1/8)sin(t)| maximum.  The || lines here mean the 'absolute value of' ignoring the sign of the answer.

We cannot do much about -1/8 as its fixed so the only chance to vary the speed comes from the sin(t) term.

Look at https://www.mathsisfun.com/algebra/trig … raphs.html

This shows the graph of a sine function.  It has a maximum positive value at t = pi/2 (measured in radians**) and a minimum at t = 3pi/2.  And it is a periodic function, meaning the values repeat every 2pi.

So there are many answers; every time a max or min occurs.  We could write the answer like this:

pi/2;  3pi/2;  5pi/2;  7pi/2 etc etc.

A neater way to say this is to give the first value, pi/2, and then give a term that describes all the repeats in one go.  The repeats are pi apart so n.pi does this (where n is any integer).  Writing the answer in this way is called the general answer taking full account or the periodic nature of the sine curve.

** I'm hoping you know about radians.  If not read on:

When you first meet angles you are taught that there are 360 degrees in a full turn.  This way of writing angles is not the only way.  In calculus d(cosx)dx = -sinx only if the angles are measured in radians. The conversion is:

pi radians = 180 degrees.

so sin(90°) = sin(pi/2) which is why the answer is given with t in  radians not degrees.

More here: https://www.mathsisfun.com/geometry/radians.html

Bob


What a great reply. Good math notes here. Thanks.

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