Last year I had a fun idea for a question on their semester exam which turned out to be even more fun than I had expected. I set up a ramp with a ball on a table, and had a trophy cup (think Goblet of Fire but smaller and plastic) on the ground a ways away from the table. Students had to figure out how far up the ramp to place the ball so that it would land exactly into the cup. I would give them all of the dimensions, the ramp angle, the mass of the ball, and the “friction coefficients”, and they would use calculations to find out how far up the ramp the ball must be placed. I put friction coefficients in quotes because it is a ball and so does not have kinetic friction. But as we haven’t gotten into rotational motion, they all know what I mean.
The students get nervous when they see the setup, and most of them want me to put the ball on the ramp. The ball is fairly small, and the cup is very small and far away from the ramp, so they have to put their trust in their math. However, the track is more lenient than it would appear. Last year there was a range of about 15 cm where you could place the ball, so many students got some fairly wrong answers and yet were successful in the experiment. This year I believe I reduced that range by increasing the angle of the ramp, and so there was only a 6 cm gap where the launch would work.
What is great about this question/practical on the exam is that I, as the teacher, can set up the experiment, and fiddle with the numbers to make the problem work. I first measure out the distances and, ideally, the angle of the ramp, and then work the calculations backwards to find the coefficient of friction. It works out great because as long as they do the math correctly, they’ll get the problem I set up, and it will work for them! This year I even fudged the angle of the ramp because I forgot to measure that while at school, and so I guessed an angle and then find the coefficient of friction based on that (and give the students my guessed angle so the math works out for them). Of course, the students don’t know that I worked everything out backwards to get the right answer, and so it feels more authentic to them.
Here’s the exam question. Since I have to set it up every year, I have to find the values myself, and so I’m not at all worried about students seeing this online and copying it from last year.
The other great thing about this problem is that the students get to see the result of their calculations immediately. They can then use what they saw (ball landed too long or too short) to make an estimate for a good “reasonable answer” and test their new final answer against that.