I've shared broadly my "double the speed of a cart" in-class laboratory exercises. This is my introduction to Bertha's Rule of Ones. In each of four situations, students use simple kinematics equations and semi-quantitative reasoning to predict a factor of change.
One exercise asks students to roll a cart down an incline, then double the cart's mass - what happens to the maximum speed of the cart on the same incline? It's easy to double the mass of a PASCO cart, just put a specially-fitted 250 g bar on top.
Another asks students to double the travel time for the cart rolling down the incline, and predict what happens to the distance traveled by the cart. Again, simple - just release the cart from rest and use frame-by-frame video. The PASCO tracks even have a centimeter scale taped on!
It seems like it would be more complicated to double a cart's acceleration, but it's not - the acceleration of a PASCO cart on an incline is gsinθ. For small angles, doubling the angle θ will also double the acceleration. Changing, say, from 6 degrees to 12 degrees gives a pretty obviously doubled acceleration.
The tough one is, how do I double the initial speed of a cart rolling up an incline?
Answer: use the plunger on the PASCO cart.
Take a look at the photo. On the plunger are three lines with numbers. (These numbers are the same color as the translucent plastic - hard to get a clear photo! I colored the lines for better visibility.) When the plunger is depressed, it clicks and stops at each of these lines. Then, a quick press of the button on top releases this plunger.
Put the plunger up against a wall, or against the stopper that can be attached to a PASCO track. Push the plunger in to position 1; push the button to release. The cart will move with some initial speed. (No clue what speed! But it will be a reasonably consistent speed each time.)
Then, push the plunger to position 2 and push the button to release. The cart will move with twice the speed as when you used position 1. And position 3 gives three times the speed of position 1.
Why does this work? Because the plunger is converting spring potential to kinetic energy. Set the spring potential formula (1/2)kx^2 equal to the kinetic energy formula (1/2)mv^2: the spring compression x has a linear relationship with the speed v.
Love the lab idea. Some teachers may want to take this further and have students solve for the spring constant of the plunger. This doesn't work well unless you know the spring is compressed about 1 cm when the plunger is fully extended. See this thread for details and a way for students to find the initial compression: https://twitter.com/kilroi22/status/1166054371114700800
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