Create Function: Height vs. Time (Bicycle Wheel)

[Adam2987]

Homework Statement

I need to create a function to represent height as a function of time in a given word problem when given the radius, constant speed and initial height. The back wheel of a bicycle is 25 cm in radius. The constant speed of the cyclist is 12 m/s. I need to calculate the height of the hind wheel from the ground at given points in seconds. The initial height is 0 (ground level). This question is really confusing me because I know what my answer is suppose to look like, but I can't achieve it. Never worked with circles before, it's usually when an item is thrown up in the air or a person jumps off a bridge into water.

Homework Equations

The Attempt at a Solution

Ok so I have a data table that looks like this.

t h
0 0
1 ?
2 ?
3 ?
4 ?
5 ?
6 ?

Ok I know that because its a wheel that at certain time intervals my equation needs to amount the same value. Like if we start at 0, the height is 0. At one second the height is 8 cm, at 2 seconds the height is 16, at 3 seconds the height is 25. Once it's reached the maximum height our time intervals after 3 should start repeating. at 4 seconds the height is 16cm, at 5 seconds the height is 8 cm and at 6 seconds the height is at ground level again or 0.

Every equation I come up with doesn't satisfy my desired results. I have a feeling it has something to do with radius and circles because I've had a fair amount of practice at this, but never a problem like this one.

Any help is very appreciated, just need a push in the right direction.

This is a Grade 12 Advanced Functions course if anyone is interested.

 

[Adam2987]

Of course this example I've provided is based on the assumption that one full rotation takes 6 seconds. This isn't at all logical, but I'm breaking it down to a simpler form.

 

[berkeman]

Homework Statement

I need to create a function to represent height as a function of time in a given word problem when given the radius, constant speed and initial height. The back wheel of a bicycle is 25 cm in radius. The constant speed of the cyclist is 12 m/s. I need to calculate the height of the hind wheel from the ground at given points in seconds. The initial height is 0 (ground level). This question is really confusing me because I know what my answer is suppose to look like, but I can't achieve it. Never worked with circles before, it's usually when an item is thrown up in the air or a person jumps off a bridge into water.

Homework Equations

The Attempt at a Solution

Ok so I have a data table that looks like this.

t h
0 0
1 ?
2 ?
3 ?
4 ?
5 ?
6 ?

Ok I know that because its a wheel that at certain time intervals my equation needs to amount the same value. Like if we start at 0, the height is 0. At one second the height is 8 cm, at 2 seconds the height is 16, at 3 seconds the height is 25. Once it's reached the maximum height our time intervals after 3 should start repeating. at 4 seconds the height is 16cm, at 5 seconds the height is 8 cm and at 6 seconds the height is at ground level again or 0.

Every equation I come up with doesn't satisfy my desired results. I have a feeling it has something to do with radius and circles because I've had a fair amount of practice at this, but never a problem like this one.

Any help is very appreciated, just need a push in the right direction.

This is a Grade 12 Advanced Functions course if anyone is interested.

There may be an easier way to do it, but I'd just use trig to write the height as a function of time.

Draw a diagram of the wheel, and several points around the wheel. Draw the radius vector to each point, and use trig (and the fact that the center of the wheel is a radius r off of the ground) to calculate the height. Also use trig and the fact that the center of the wheel has a constant velocity forward, to calculate the horizontal position of the point. That all gives you the x,y position of the point on the wheel as a function of time.

Post some tries at that if you are still having trouble.

 

[Adam2987]

I'm not sure if trig is the best way to go about this. This question is in the 4th unit (Mathematical Modelling), Trig was the 3rd unit. The units don't usually build upon one another, just the lessons within each respective unit. I'm not saying that trig can't solve this, I'm just not sure if I'll get marks when I show how I came up with the equation. An example of the equations I've been given so far in this unit is h(t) = -5t^2 +30t. That was an equation I could use to calculate a diver jumping off a bridge, and determining how far from the bridge or how close to the water he was. I'm assuming I need to do something similar. In that example equation you just had to plug seconds in, and it gave you the appropriate response in height. Seeing how there was absolutley no trig in this unit, I'm just weary if I can actually find the equation for this problem using it.

 

[Adam2987]

I don't need to solve for height at each second. The question only states that I come up with an equation to determine height. So merely solving for x and y isn't going to net me any marks. And every equation I am creating is giving me whacky results. I'm just wondering if there is some trick to these types of problems.

 

[Gigasoft]

This is actually a trick question. Assuming that the cyclist is always moving in a fixed horizontal direction along a perfectly horizontal ground, and the bicycle is comoving with the cyclist (being at rest relative to the cyclist), it is easy to see that the function can be written as f(t) = 0. The vertical component of the cyclist's velocity, and therefore of the bicycle's velocity, is 0. We assume that the surface of the wheel has circular symmetry about a line coming through its center that is perpendicular to both the direction of movement and the vertical axis, meaning that the wheel is not turning or leaning to the side, and that the wheel is also rotating about the same axis, where the center is at rest relative to the bicycle. Then it can be shown that the shape of the surface of the wheel remains constant at all points in time as seen from the bicycle's reference frame. Since the vertical velocity of the wheel's center point is zero, its height is a constant, and the wheel's height above ground (the minimum distance in the vertical direction from any point on the wheel to the ground) must also be a constant. It is therefore equal to the initial height, which is 0.

 

[Adam2987]

That makes a lot of sense haha. No wonder I couldn't figure out an equation.

 

[Adam2987]

But, if we take a certain point on the wheel, (we'll call it x) at such a time let's say 1 second, assuming the tire hasn't made a full rotation, is it not higher than its original position?

At 0 the initial height is 0

At 1 second, our point of x should be higher than 0 assuming the bike is in motion right? ( and assuming it has yet to make a full revolution)

 

[berkeman]

This is actually a trick question. Assuming that the cyclist is always moving in a fixed horizontal direction along a perfectly horizontal ground, and the bicycle is comoving with the cyclist (being at rest relative to the cyclist), it is easy to see that the function can be written as f(t) = 0.

No, I believe the question is to determine the equations for x(t) and y(t) for a single point on the tire that is initially at y=0 at t=0. As the tire spins with the bycycle's motion, that single point will move up and down and also in the direction of the bicycle's motion.

@Adam2987: A single point on a wheel describes a particular shape as the wheel rolls. I forget the name of the shape, but will see if I can find it. Once we have that name, you can use wikipedia or some other source to help you with the equations.

 

[berkeman]

Well, that wasn't such a chore now, was it. (Quiz Question -- what movie is that line from?)

I googled point on a rolling wheel. Try that, and show us what you come up with. Have fun!

 

[Adam2987]

http://qbx6.ltu.edu/s_schneider/physlets/main/rollingwheel.shtml

Here's the website I found. Looks like a sinusoidal wave. Thanks for the pointers. I'm going to try and work soemthing out right now.

 

[Adam2987]

Well, that wasn't such a chore now, was it. (Quiz Question -- what movie is that line from?)

The movie is Ghostbusters by the way...lol

 

[berkeman]

The movie is Ghostbusters by the way...lol

Ding Ding! Glad you're back on track.

 

[Adam2987]

Ok so after some toying around, I'm thinking I need to derive a sin or cos equation using amplitude, period and 1/4 wave but I can't seem to determine these and the horizontal axis. Yikes!

 

[berkeman]

Ok so after some toying around, I'm thinking I need to derive a sin or cos equation using amplitude, period and 1/4 wave but I can't seem to determine these and the horizontal axis. Yikes!

Post your work so far. You know the PF mantra!

 

[Adam2987]

f( x) = 25cos(3pie/6)
25 is the amplitude. 2pie is the period, 1/4 wave is pie/2
There is no phase shift or vertical shift. There is a reflection in the x-axis I believe though.

This is what I have so far. Working on the rest or stabilizing the equation still though. With a lot of difficulties.

 

[Adam2987]

Ok here's what I've come up with
25cos(pie/3x)+12

substitude time for x and you get your height.
How does that look. It seems to give an appropriate answer

 

[Adam2987]

Although when I sub x=0 I don't get 0, which is misleading from the question. Gahhhh. Back to work I suppose.

 

[Adam2987]

25cos(pie/3t)+12t

I think I got it lol

 

[Gigasoft]

Here are some issues to think about. Think carefully about each question in order and move on when you think you are sure of the answer.

$$f\left(t\right)=25cos\left(\frac{\pi e}{3t}\right)+12t$$
This is what you mean, right? Can a cyclist ride straight upwards at 12 m/s, or 43.2 km/h? Are typical bike wheels 50 meters in diameter? Should f(t) be defined when t=0? Is it defined? Where do $$\pi$$ and $$e$$ come from?

Have you tried graphing your function? Go to http://www.walterzorn.com/grapher/grapher_e.htm", enter the formula 25*cos(pi*e/(3*x))+12*x, plot the function with x ranging from -10 to 10 and y ranging from -200 to 200. Is this a reasonable path to take vertically for the hind wheel of a bike, or a point on the wheel, during a typical bike ride?

If you assume that the question should really be about the height of a single point on the wheel, you should write this assumption as part of your answer, e.g. "Assuming that the question came out wrong and that the author really meant..."

If that's the question you think you need to solve, then what path does a fixed point on a circle take when the circle rotates about its centre? If the bike is moving forwards at a constant speed, what can be said of the rotation speed of the wheel, and what can be said of the path of a point on the wheel as seen from an observer in the centre of the wheel? If you roll a wheel along the ground, how does the point on the wheel that touches the ground, move relative to the ground? What is then the velocity vector of that point on the wheel relative to a bike moving forward at 12 m/s? What is the speed relative to the center of the wheel? What is the magnitude of the speed of any other point on the edge of the wheel, as seen from its center? (Hint: How does the distance between neighbouring points on the edge of the wheel change as the wheel rotates?) Knowing the point's speed along the path it takes at every point along its path, and the length of the path it takes when starting from the bottom of the wheel, going around it and back to its starting point, what is then the time it takes to complete one revolution? If the cos function is to repeat after the time a revolution takes, what must its argument be? For which argument does the shape of the cos function resemble the path of a point on the wheel which is initially at the bottom? Taking this into account, what must its argument be then? What is the function's value at t=0, and what must you do to make it fit the boundary conditions while retaining its shape? Now we add in the horizontal speed of the bike. How does this affect the vertical component of the speed of each point on the wheel? If you know the answer to all of this, you should have no problem writing the correct answer.