Finding Magnitude of Velocity: Using Differentiation and Implicit Equations

[Sheen91]

Homework Statement

Given that:

8y^{2} + 4y = -4x + 6
x + y = 4t

where x and y are in metres, and t is in seconds.

If y is positive, what is the magnitude of the velocity when t = 0.1875s?

Homework Equations

N/A

The Attempt at a Solution

I am a little unsure on how to start.

I think I have to differentiate:

8y^{2} + 4y = -4x + 6

Using implicit, then sub in x + y = 4t

And solve for x and y.

If all that is right, which I doubt it is. I then have to some how find the magnitude of velocity, which I am unsure how to do.

Any and all help would be greatly appreciated. :D

 

[Raskolnikov]

You're pretty close.

Use implicit differentiation on the first equation, and solve for y' in terms of x and y.

Also, solve for y in the first equation when x = 4(0.1875) - 7. Take the positive root of y since the problem says y > 0.

Now plug in the positive root you found for y and x = 4(0.1875) - 7 into the equation you found for y' in the first step. This should be the velocity at t = 0.1875 s.

 

[Mandeep Deka]

what i basically see is that u have been given the equation of the trajectory of a particle and the relation between its x co-ordinate ad time is given.
So what u do is:

obtain, dx/dt=4, which is the velocity along x-axis,
at t=0.1875s, u can find x from the 2nd equation and hence plug its value into the 1st equation to obtain y. Once u have the x and y values of the function at t=0.1875s,
differentiate the 1st equation u wrote, with respect to 't',
u will get dy/dt in terms of y and dx/dt. hence, dy/dt is the velocity along y axis.
at t=0.1875s, plug the value of y u obtained, and u have already calculated dx/dt, so u can obtain dy/dt.

So the magnitude of velocity of the particle at that instant is just {(dy/dt)^2+(dx/dt)^2]^0.5

 

[Sheen91]

First of I must say, sorry that the equation was x + y = 4t. My bad I accidentally wrote x + 7 for some reason. T.TOk so first I implicit differentiate:

8y^{2} + 4y = -4x + 616y . \frac{dy}{dx} + 4 . \frac{dy}{dx} = -4\frac{dy}{dx} = \frac{-4}{16y + 4}

Ok so now I solve for y in the second equation:

x + y = 4t = \frac{3}{4} = 0.75

y = 0.75 - xNow I am kinda stuck again. Because I am unsure what to do. If you sub in that equation you will always have one variable in it. Not sure what to do.

 

[Dick]

I would split this into two parts. First put your value for t in. Now you have two equations in two unknowns for x and y. Find them. Then differentiate d/dt. Now you have two equations in two unknowns for x'=dx/dt and y'=dy/dt. Use the value you found for y in the first part. Then the magnitude of the velocity is sqrt(x'^2+y'^2). I haven't tried to work it out, but does that make sense?

 

[Sheen91]

x = 4t - y

dx/dt = 4 (because then y would be a contant.)

You want me to sub in x + y = 4t into the equation below?...?...

8y² + 4y = -4x + 6

8y² + 4y = -4(4t - y) + 6

8y² + 4y = -16t + 4y + 6

8y² = -16t + 6

y = sqrt(-2t + 6/8)

dy/dt = 0.5 x -2 / sqrt(-2t + 6/8) = - 1/sqrt(-2t + 6/8)

I guess I will sub in t = 0.1875, and I get 1.63, square that and 4, and root it. I get 4.320 m/s. Which unfortunately is not right.

 

[Dick]

These aren't partial derivatives. y isn't constant. It's a function of t. Differentiating x=4t-y gives you dx/dt=4-dy/dt.

 

[Sheen91]

Ok so

8y² + 4y = -4x + 6

16y dy/dt + 4 dy/dt = -4 dx/dt

(16y + 4) dy/dt = -4 . dx/dt

sub in dx/dt = 4 - dy/dt

(16y + 4) dy/dt = -4 (4 - dy/dt)

(16y + 4) dy/dt = -16 + 4 dy/dt

(16y) dy/dt = -16

(y) dy/dt = -1

dy/dt = -(1 / y)

And

dx/dt = 4 - dy/dt

dx/dt = 4 + (1 / y)

Now I we have dy/dt and dx/dt, I hope. :S

I am unsure how to get the velocity?

I know velocity = sqrt(x'^2+y'^2)

but there is still the y in the equation, and I can't remove it using "x + y = 4t"

because that just changes y into x, and vise versa, and you still get an x or a y in your answer.

 

[vela]

Close, but you made a sign error right at the start when you differentiated -4x. Try fixing that first.

 

[Sheen91]

fixed

I am still a little stuck at the same point

 

[vela]

If you can figure out what y equals at t=0.1875 s, you can calculate the two components. So following Dick's suggestion, set t=0.1875 s in the original equations, and you'll get

8 y^2 + 4y = -4x + 6

x + y = 0.75

You have two equations and two unknowns. Solve for y.

 

[Sheen91]

8y² + 4y = -4x + 6

sub in : "x = 0.75 - y"

8y² + 4y = -4(0.75 - y) + 6

8y² + 4y = -3 + 4y + 6

8y² = 3

y = sqrt(3/8) - because y > 0

If I sub that value into dx/dt and dy/dt

dy/dt = -(1 / y)

dy/dt = -(1 / sqrt(3/8)) = -1.633

dx/dt = 4 + (1 / y)

dx/dt = 4 + (1 / sqrt(3/8)) = 5.633

velocity = sqrt( 5.633² + (-1.633)² ) = 5.865 m/s

WHICH IS CORRECT! Thank you so so much :D