Physics: ball launch using gravity

[JoeXIII'007]

OK, before I get into what I'm looking for, here's my assignment:

I have to build a device that will launch a ball approximately 10 grams in weight to a 10 cm wide hoop 1 meter away and 1 meter high.

Now, what I really want to do is build a ramp system that will use gravity to accelerate the ball and then redirect its direction so that it will launch high and far enough to get into the hoop. In my thinking, if I build it just right, it should work 100% of the time.

Question is, how do I successfully transfer the vertical motion of the ball at start into horizontal and vertical motion good enough to get the ball into the hoop?

Thoughts?
Other suggestions are really welcome.

-Joe

[edit] in the case that this involves complex math and you must use it, I'm currently enrolled in a pre-calc class. So go for it. [/edit]

[David Brinza]

Using conservation of energy, you should be able to determine from what height you will need to start rolling the ball down a "ski jump" in order to achieve enough velocity to launch the ball at some angle to make it through the hoop. Don't forget to include the rotational energy of the ball in your calculation...

[Tom Bottiglieri]

Originally Posted by David Brinza

Using conservation of energy, you should be able to determine from what height you will need to start rolling the ball down a "ski jump" in order to achieve enough velocity to launch the ball at some angle to make it through the hoop. Don't forget to include the rotational energy of the ball in your calculation...

As David said...

Total energy is a system is constant.

Potential Energy (U)  = (mass) * (height) * (acceleration of gravity)
Linear Kinetic Energy (T) = 1/2 * (mass) * (velocity ^ 2)
Rotational Kinetic Energy (KR) = 1/2 * (moment of inertia) * (velocity ^ 2)

I for a sphere: 2/5 * Mass * Radius^2

EA = EB (E is total energy, A and B are arbitrary points in time.)

E = T + U + KR

So what I would do first is a simple kinematics/projectile motion problem (there might be some handy tools in the white paper section ) to find what velocity is needed on the launch. (In this case, you would arbitrarily chose the angle theta of the launch. I would chose 45 degrees.)

Once you have your needed velocity and launch angle, figure out how much kinetic energy it will have at launch. Plug the velocity, mass, and Moment of Inertia (I) into the equations above, and add the 2 values together to find net kinetic energy. Then, set that equal to the potential energy (You can assume this if your reference frame has the balls launch point as (0,0) and the ball is starting from a stop, aka no kinetic energy.)

So now you have U = T + K_R and U = mgh. Simply solve for U, and divide by 0.098 (g * m) to find how high (in meters) your ball should start in the y direction above the launch point.

Voila.

[JoeXIII'007]

Originally Posted by Tom Bottiglieri

As David said...

Total energy is a system is constant.

Potential Energy (U)  = (mass) * (height) * (acceleration of gravity)
Linear Kinetic Energy (T) = 1/2 * (mass) * (velocity ^ 2)
Rotational Kinetic Energy (KR) = 1/2 * (moment of inertia) * (velocity ^ 2)

I for a sphere: 2/5 * Mass * Radius^2

EA = EB (E is total energy, A and B are arbitrary points in time.)

E = T + U + KR

So what I would do first is a simple kinematics/projectile motion problem (there might be some handy tools in the white paper section ) to find what velocity is needed on the launch. (In this case, you would arbitrarily chose the angle theta of the launch. I would chose 45 degrees.)

Once you have your needed velocity and launch angle, figure out how much kinetic energy it will have at launch. Plug the velocity, mass, and Moment of Inertia (I) into the equations above, and add the 2 values together to find net kinetic energy. Then, set that equal to the potential energy (You can assume this if your reference frame has the balls launch point as (0,0) and the ball is starting from a stop, aka no kinetic energy.)

So now you have U = T + K_R and U = mgh. Simply solve for U, and divide by 0.098 (g * m) to find how high (in meters) your ball should start in the y direction above the launch point.

Voila.

Thank you. This should work really well.

-Joe

[David Brinza]

Originally Posted by Tom Bottiglieri

So what I would do first is a simple kinematics/projectile motion problem (there might be some handy tools in the white paper section ) to find what velocity is needed on the launch. (In this case, you would arbitrarily chose the angle theta of the launch. I would chose 45 degrees.)

I posted an Excel spreadsheet that computes the ball trajectory for Aim High. You can adjust the launch angle, velocity, and see the effect of air resistance (which should be really small) for your problem. To "turn off" air resistance, you can set the drag coefficient (Cd) to zero. The spreadsheet can be downloaded from this post:

http://www.chiefdelphi.com/forums/sh...5&postcount=19

Have fun...

[JoeXIII'007]

It's been a while since I posted here, but a few things have changed with the project, which is working quite well. First, here's the criteria for the ball launcher from the sheet my physics teacher gave:

Build a machine that will project a ball (current weight is roughly 30-40 grams, it is no longer 10) through a hoop that is ~ 10 cm wide, 1 meter in the x direction, and 1 meter in the y direction.

Rules:

• you will have three tries to get the ball through the hoop.
• The ball must move at least 1 meter in the +x direction once it leaves the bounds of the machine.
• Calculations to support your design must be completed in report form. Include the following in the calculations:
  • Force of the mechanism acting on the ball.
  • Graph showing the path of the ball once it leaves the mechanism.
  • Speed the ball is traveling through the air

Grades will be determined by the accuracy of calculations and physics

• 50pts Mechanics of the machine (does it work)
• 35pts Creativity of the design
• 15pts Extra points may be earned for distances traveled through the air > 1 meter.

So, here's the status of the project, which is due next Wednesday. I have it launching the ball 10-15 cm short of the 1 meter height and 1 meter away requirement. HOWEVER, I have been able to consistently get the ball through the hoop, which is in vertical orientation, between 84-94 cm high. The angle, which I am yet to measure is about 45-55 degrees. Do not trust the protractor like device in one of the pics, for I do have the ramp a bit higher than 45.

The device consists of a vertically standing PVC pipe that is 1.9 meters long. It provides the ball gravity acceleration. At the end of the pipe, I have 2 'hot-wheels' flexible tracks that end about .75 horizontal meters away, pointing the ball 45-55 degrees to the hoop.

I need to know how exactly to measure the curve, circular or not. It was never drafted, and was tweaked SEVERAL times that any rough/initial draft with any accuracy would be no longer accurate. Pictures are attached of the device, or should I say ramp.

As far as calculations are concerned, I know that I can get the speed and force before the ball hits the curve using v = gt and f = ma respectively since it's a relatively straight vertical drop. It's the curve that I need measured, and the angle I can get easily.

Thanks again, and any ideas on how to get the ball just a wee bit higher would be really great.

-Joe

[David Brinza]

Joe:

Does your calculation of drop height consider the rotational kinetic energy of the ball? (If not, this would explain why you are coming up short.)

You may also be losing some energy due to flexure of those "hot wheels" tracks and perhaps some friction losses as well. So, you will need to drop the ball from a greater height to offset some of these loss factors. If you need the ball to go higher, you can increase the angle (with loss in distance traveled).

[Richard Wallace]

Originally Posted by JoeXIII'007

Thanks again, and any ideas on how to get the ball just a wee bit higher would be really great.

-Joe

This may be a bit late in your project and a bit too abstract to be useful, but..

Neglecting friction, the problem of finding the curve of steepest descent rate (which would maximize velocity at the bottom) is called the brachistochrone problem. The solution is a cycloid curve.

The problem can be generalized using calculus of variations to include friction, rotational KE, etc.

The descending section of your Hot Wheels track appears to be very close to a cycloid.

[JoeXIII'007]

Originally Posted by David Brinza

Joe:

Does your calculation of drop height consider the rotational kinetic energy of the ball? (If not, this would explain why you are coming up short.)

You may also be losing some energy due to flexure of those "hot wheels" tracks and perhaps some friction losses as well. So, you will need to drop the ball from a greater height to offset some of these loss factors. If you need the ball to go higher, you can increase the angle (with loss in distance traveled).

Actually, the 1.9 meter length of the pipe was arbitrary, because I bought over at a hardware store, and they cut it there. I do have the other half, however. I have actually gone the experimental route in developing the launcher.

Increasing the angle I have tried, but the loss of x-distance is too great.

And yes, I have been trying to stiffen the tracks so that they would not absorb energy needed to launch. As far as height, we have considered going to the second floor and drilling a hole large enough to do that, but I do not think the administration and especially the custodial staff would like that. The top of the pipe is 10-15 cm from the drop ceiling though, and perhaps if I could convince someone to let me move the tiles... hmm... yes. I'll have to do some good old fashined negotiating.

Originally Posted by Richard

This may be a bit late in your project and a bit too abstract to be useful, but..

Neglecting friction, the problem of finding the curve of steepest descent rate (which would maximize velocity at the bottom) is called the brachistochrone problem. The solution is a cycloid curve.

The problem can be generalized using calculus of variations to include friction, rotational KE, etc.

The descending section of your Hot Wheels track appears to be very close to a cycloid.

I'm studying Pre-calc, not calculus quite yet, that's a problem. But if I were to sit down and take some time with it, maybe I could figure it out. Otherwise I do not need to bother my physics teacher with more stuff to read over, since he was assuming everyone was going to build a simple catapault or trebuchet. Plus, he's got a hockey team, robotics team, TV productions, and this physics class to handle (a lot of people need help on simple physics stuff, I always try to help out). SO, I'll have to ask him what exactly he would like as far as calculations for the device.

Interesting stuff though, can't wait until I learn it all, and thank you very very much for the insight.

-Joe

PS: I wonder what my partner in this project is going to think of all this...