Challenge: To design an adjustable, calibrated ramp that will make a 188 gram cart with a 500 gram passenger and provided flag achieve a target velocity at a specific target location, when released from rest.
Parameters: Your ramp must be at least 50 cm long, and the target location will be between 20.0 cm and 40.0 cm away from your starting position, which you will find out just before the competition. The flag will have its width written on the top, and it must be 15 cm from the surface of the ramp. The velocity asked of you will be between 0.750 m/s and 2.250 m/s. You will get up to three trials, but all will count for scoring purposes.
Scoring: Your system will be scored based on the absolute percent difference between the target velocity and the actual velocity of your cart, and the number of tries.
Score = (1.1)(n-1) | Your velocity - Target velocity |/ Target velocity *100%
Where "n" is the number of tries. Lowest score will receive highest rank.
Physics: The cart will accelerate down the incline according to Newton’s second law. The component of the force of gravity acting down the incline will be the “winner” while the frictional forces will be the “loser”. The velocity “v” at a displacement “d” can be predicted from the acceleration by using the equations of constant acceleration. (v2 - v02 = 2ad where the initial velocity is zero). You will need to determine how the acceleration relates to the ramp angle for your system. You can do it mathematically if you have a good estimate of the friction forces or you can calibrate your ramp by measuring actual acceleration at various angles and finding the equation of the trend line with a spreadsheet program.
Extensions: The flag/photogate method finds an average velocity over a short interval, which we are using as a measure of instantaneous velocity at the position of the photogate. This introduces some error. See if you can account for and correct for that error. Rolling friction does not perfectly follow a simple relationship like Ff = µFN and the wheels have something called rotational inertia, which means the acceleration is not exactly constant during the trip down the ramp. The calibration method will only work if you have lots of quality data to put in the spreadsheet.
Help/Hints: Try to use a “good” cart, and a straight ramp, and make sure your flag is perfectly vertical. Put a piece of tape on the bottom of your cart with your table code on it so you can use the same cart each class. You may design your own cart and mass system, but the flag must be at the correct height to be recorded.
Quiz Topics: Kinematics and Dynamics in two dimensions. (Newton’s Laws of Motion)
Online Text: ch 4, ch 5, ch 6