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## 13 December 2010

### Two-cart elastic collision -- how to measure the speed of both carts

I've taught conservation of momentum with the same set of quantitative demonstrations for years. I get two carts and a motion detector, make the carts collide, and predict speeds of the carts before or after the collision. Pretty basic, without much room for creativity. Until today, that is.

The first collision problem I attempt is very straightforward.  Take a look at the setup in the picture to the right.  I keep the red cart at rest, and send the heavier blue cart toward the red cart.  A velcro strip on the carts causes them to stick together after the collision.  The motion detector measures the speed of the blue cart before and after the collision.  I tell the class what the motion detector said the blue cart's initial speed was; we predict the speed of the combined carts after the collision.  Generally we get this prediction correct within 5-10%.

In the next collision, I push the blue cart toward the stationary red cart without the velcro, so that the carts bounce off one another.  Again the motion detector records the speed of the blue cart before and after the collision; we predict the speed of the red cart after collision.  Problem is, how should I measure the speed of the red cart to verify my prediction?  I pose that question to the class.

Most quickly understand why the single motion detector can only read the blue cart's speed.  The detector can't "see through" the blue cart.

The solution suggested by a majority of students is to place a second motion detector on the left side of the track to track the red cart.  And I've tried that before.  But I've never gotten two facing detectors to work properly -- I think the sound waves interfere, causing nonsense results.  (Anyone else have any thoughts on this issue?)

What I've always ended up doing -- until today, anyway -- is to have students measure the time between the collision and when the red cart hits the end of the track, 100 cm away.  Since the track is level, the speed of the red cart is 100 cm divided by the time the students measure.  This works, but stopwatch measurement uncertainty causes this to be accurate only to about 20-30%, if that.

In today's class, junior Will Choate made the suggestion that I've missed for the last 15 years:  "Can you pick up the blue cart after the collision?  Then the detector can read the red cart."  That's IT!

So, I collided the carts.  After the collision, I gave the detector a brief moment to read the blue cart's speed, then I picked up the blue cart.  Sure enough, the velocity time graph showed the initial speed of the blue cart (47cm/s); the final speed of the blue cart (31 cm/s); a bunch of nonsense where my hand briefly got in the way; and the final speed of the red cart (80 cm/s).  We used conservation of momentum to predict that the red cart should have been moving 84 cm/s, within 5% of the measured speed.

The class was pleased with Mr. Choate for his creativity, and because he earned a TWO Reese's Cup reward.  I think much of the applause and congratulations was kissing up to Mr. Choate in case he felt inclined to give away his candy, but nevertheless.