What is the Solution for Finding Velocity and Distance on a Water Slide?

[Clever_name]

Homework Statement

Hi all, I am currently working on a problem that i believe involves the work-kinetic energy theorem.
See the attachment and information below.

As shown in the figure, the geometry of the water slide used is 1m:
Slide length = 61,; Slope angle = 24; Ramp height = 3.66 m

Lubricant is used thus friction can be ignored. A man of 70kg slides from the start.
if the ramp angle is 30, find the magnitude and direction of the velocity of this man when he just flies away from the ram and the Horizontal distance that the man flies.

Homework Equations

W=(1/2)mvf^(2) - (1/2)mvi^(2)

The Attempt at a Solution

ok so first I find the force that is acting on the man as he starts sliding down the ramp. which comes out to be mgcos(24) = F

and i know the distance of the slide which is 200ft. And further we can say W=FD
so subbing values in and solving for vf I get 108.43 ft/s - this is the velocity when the man is at the the lowest point on the slide.

So then i find i repeat the process and find vf at the top of the ramp with an angle of 30. which gives me an answer of 102.08 ft/s.

There is no answer to this question so i have no idea if I'm right or not... any help would be greatly appreciated.

also to find the direction of this velocity can i assume at the instant he comes off the slide then the angle the slide makes to the horizontal is also the direction of his velocity?

Thank you again!

 

[Clever_name]

disregard my initial workings, i used cosine when i should have used sine... still not sure if my answers are correct however.

My new answer is 66.8 ft/s when the man leaves the ramp.

 

[Simon Bridge]

I'm afraid that docx file won't come out on my computer.
Please do not provide important information in files that contain proprietary formatting and/or extensions - not everyone can open them.

From your description, this is a conservation of energy problem but you don't need the work-energy theorem.
Start by describing the motion of the man in terms of the energy changes that occur in each stage.

 

[Clever_name]

Hi Simon! Try this link http://i49.tinypic.com/2vdh3yf.jpg to view a picture of the situation.

 

[Simon Bridge]

OK - now I get it.

According to that diagram, the distance from the top of the slide to the point on the slope directly below the top of the ramp is 200 feet. The man does not go down a 200 foot slide and then up a ramp ... the ramp is part of the 200 feet and the overall length of the (colored brown) surface is going to be greater than 200 feet overall.

So the work-energy relation is not going to help you here - the force you worked out does not act for the entire 200 feet. *

My suggestion at post #3 stands: describe the motion in terms of energy transformations and use conservation of energy.

What sort of energy is lost going from the top of the slide to the top of the ramp?
What does it turn into?

---------------------

[*] actually the arrow indicating the 200 foot length stops a bit short - yet goes past the curve to the bottom of the ramp ... so it is a little ambiguous. You could check what is intended here - but, if you use conservation of energy as suggested, it won't matter ;)

 

[Clever_name]

Ok so my new attempt is as follows:

mgh = (1/2)*mv^(2)

h= 69.35 ft - subbing this into the equation and solving for v i get 66.8 ft/s.
Not terribly confident however that this is correct because its the same result i got using an incorrect method.

 

[Simon Bridge]

You left out a lot of steps there so I cannot see how you got those numbers.

You did: h=200(ft)sin(24°)-12(feet)=69.347 ft
... with g=33ft/s/s, v=√(2x33x69.347) = 67.653ft/s

Perhaps several errors canceled out?
Perhaps I misread what you did?
Whatever - can you see how this way was more direct?

 

[Clever_name]

yes, however there is a second part to the question which is also troubling me. the question also requires me to find the direction of this velocity when he leaves that ramp. can i assume it will be the same angle that the ramp makes with the horizontal i.e 30?

I will need this value to find out the horizontal distance that the man flies when he leaves the ramp.

 

[Simon Bridge]

That's correct.
After that - finding the horizontal distance is normal ballistics.

 

[Clever_name]

Hi Simon - sorry to pester you again but can i get you to verify my answer as there is no solution i have no idea if my working is correct :(.

Ok so my solution to find the horizontal distance is as follows:

finding the final y component of velocity
Vfy^(2) =Viy^(2) +2a*Δy
I have the variables to solve for Vf, since Vi^(2) = 66.8sin(30), g = -32.174 and Δy = -12
so Vf = 43.44 ft/s

Now i use this to find time.

Vfy = Viy +at

So Vfy = 43.44 Viy = 66.8sin(30) and a = -32.174

so t = 2.38s

and thus Δx can be found.

Δx = Vix * T = 66.8cos(30)*2.38 = 137.683 ft

Thanks again!

 

[Clever_name]

Ok so the final part of the question is as follows:

Use theoretical methods to determine the magnitude of the ramp angle that will result in the max flying distance.

Honestly I don't know where to start with this question. If i try and relate the potential energy to the kinetic energy the terms in the equation don't involve the ramp angle and thus is unhelpful. any tips would be greatly appreciated.

Thanks!

 

[Simon Bridge]

In #10 you did it in several steps using several equations - use algebra to combine the equations until you have only the initial velocity, gravity, and an angle.

I think your teacher is trying to get you out of the habit of thinking in terms of equations you have memorized and start you thinking in terms of constructing the equations you need.

 

[Clever_name]

Ok so,

Δx = Vocos(∅)t

Δy = Vosin(∅)t -(1/2)gt^(2)

since Δy =0, we can solve for t and sub it into the first equation and get an equation in terms of gravity, intial velocity and the angle.

doing this yields (2Vo^(2)sin(2∅)) / g

and this is the expression i need to maximize?

 

[Clever_name]

actually disregard that i made a mistake

 

[Simon Bridge]

Yep - i.e. $\Delta y \neq 0$
But I suspect you already know the angle that maximizes the flying distance (with no air resistance) - you just have to prove it.

It helps to simplify the notation ... let the initial speed be v so that the horizontal speed is $v_x=v\cos\theta$
Let the horizontal distance traveled by d so that $d=v_xt$ where t is the time traveled. The vertical distance traveled is the height of the ramp - h.

The initial vertical speed is $v_y=v\sin\theta$

This let's you avoid all those subscripts and deltas that can mess with your algebra.
(i.e. $h^2$ is so much nicer than $(\Delta y)^2$ or having to mess with $v_{fy}$ and $v_{fx}$ and so on.)

It is also useful to start using latex to write out equations ... the soner you start the aaseir it will be leter when you have to. For the stuff you are doing, it is pretty intuitive. That last equation, for eg, was just v_y=v\sin\theta

Some things to check:
by "flying distance" do they mean "the horizontal distance flown" ... it could mean "the length of the path taken through the air".

 

[Clever_name]

Ok I'm getting confused when trying to solve for time t.

Δy =-16, so

Vosin(∅)t + (1/2)(g)t^(2) +16

but when solving for t how do i know which one to use resulting from the quadratic equation?

if i use (-Vosin(∅)/g + (Vosin(∅))^(2) -4(1/2)(g)(16))^(1/2)/g i get the max angle at 45,

but when i use (-Vosin(∅)/g + (Vosin(∅))^(2) -4(1/2)(g)(16))^(1/2)/g and compute angle 45 compared to 46 i get a higher result for 46 than 45.

Thanks for the advice.

 

[Simon Bridge]

For your notation with "up" being positive, then the acceleration of gravity is negative.
When you are solving for t from the quadratic equation, one of them will be clearly nonsense.
You could keep the $\pm$ to the end if you like, this pick the distance that is physically possible.

 

[Clever_name]

How can i calculate the distance though, without already assuming the angle's value, which also leads me to the question of how can i tell which t is nonsense, without knowing Vo and the angle theta?

 

[Simon Bridge]

Without the initial speed and angle you cannot do this no. However, you worked out what these were in post #6 and post #8.

 

[Clever_name]

That was for the case where the angle was 30 i.e the angle was given.

 

[Simon Bridge]

Well the initial speed is the same ... you will leave the angle as a variable.
You know what range of values it should have.

You want to end up with an equation which has a distance as a function of angle.

 

[Clever_name]

ah i see, i was thinking that if i change the angle then h changes. Thanks for clearing that up Simon.

 

[Clever_name]

mmm still getting confused with this question. So i have x as a function of theta, but I'm getting 44 degrees to give me higher distance then 45. Am i meant to take the derivative of this function and set it to zero then solving for theta?

 

[Clever_name]

I used wolfram alpha to graph the x v theta graph and its giving me the maximum at 41.79. how do i find this value analytically though, thanks!

 

[Simon Bridge]

If the function has a maximum anywhere, then the slope of the function will be zero at that maximum. You exploit that property to find the maximum.

Notice that 45deg gives the maximum distance when the final height is the same as the initial height. if T is the time to return to the initial height, then v_y(T)=-vsub]y[/sub](0), while v_x(T)=v_x(0).

But, in this case, after reaching the initial height again, there is still some more distance to fall in the y direction. The time to fall the rest of the way is fixed. Therefore the extra horizontal distance traveled is determined by the horizontal velocity alone. The bigger that is the farther the distance traveled - and the shallower the initial angle. So it is reasonable to expect that the angle for maximum distance is a bit less than 45deg.

Notice that if you already know how the distance to the point where the height is the same as the initial height relates to angle, then you only have a simple step to go.

 

[Clever_name]

Are you saying i need to take the derivative of the function? because even if i do, i do not have the required math knowledge to solve it at x'(theta) =0.

 

[Simon Bridge]

That is unfortunate - you need to acquire the math knowledge.
It is as simple as that.

Trig IDs are useful here:
http://en.wikipedia.org/wiki/List_of_trigonometric_identities
i.e. $2\sin\theta\cos\theta = \sin 2\theta$

 

[Clever_name]

I'll give it a go, thanks Simon for the help!

 

[Clever_name]

Ok well I've given up on trying to solve this analytically. Just one last question - I'm using wolfram alpha to get the final answer which they give as, "theta ≈ 0.74559 + 6.28319n for integer n".

can someone tell me what this statement means.

do i assume theta in degrees equals 0.74559(180/pi)?

Thanks for the help!

 

[Simon Bridge]

There are an infinite number of values of theta which make the expression true.
Try dividing by 2π to see where the angles lie. Is there any restriction on the possible values of the angle in the problem?

Mathematica (used by Wolfram Alpha) likes theta in radians - yes.

If you get confused you can check just by plugging the number back into the expression and see what you get.

 

[Clever_name]

the restriction on the possible angle would i assume range from 0-90, or 0 - pi/2?

I don't get what you're saying when you say divide by 2n.

I divide theta ≈ 0.74559 + 6.28319n by 2n? I don't see how that would help me see where the angle lies.

 

[Simon Bridge]

Not 2n, 2π - spot the difference.
Oh OK, I'll write more clearly:
Divide by $2\pi$.

To see why: what is $2\pi$ in decimal to oh... 5dp?

Why would there be more than one correct value for the angle?
Is the value periodic? Is so, what is the period?

Whatever - do not guess about the range of the allowable angle - use the problem statement to tell you.

 

[Clever_name]

2pi to 5 decimal points is 6.283185.

and the period is 6.283185, so the period is 2pi!

I'm looking at the problem statement and cannot see why the max allowable range wouldn't be 0-90, given that we are looking for the max change in distance x.

 

[Simon Bridge]

Sure - I directed you to check because you said you "assumed".
Don't "assume" unless donkeys are involved ;)

The ramp could, technically, have been built to any angle.
If the ramp angle were between 90 and 180, for eg., you'd get a long distance but in the opposite direction... so the displacement would be negative. (And it would have hit the hill before that distance was reached.)

What happens when the ramp angle is between 180 and 360? When would the next maximum be after 360 degrees - what does the ramp look like compared with the initial angles less than 90?

In this way the physics will tell you which of your available numbers make sense.
You could arrive at the same conclusion by maths, when you remember how many radians go in a full circle.

Note: you should start using radians by default, particularly for geometry and physical sciences.
It makes things a lot easier in the long run.

 

[SammyS]

Hi Simon! Try this link http://i49.tinypic.com/2vdh3yf.jpg to view a picture of the situation.

Did you know that you can display this image directly in your post?