Force and Motion: Creating Two Classic Physics Questions in Real Life
There are two questions that can really give some insight into complicated physics concepts. However, I want to do more than just ask the questions. I want to SHOW these situations using the Vernier Fan Carts.
OK, let me start off with the easier of the two questions.
Same Force for the Same Time
Imagine I have two carts on frictionless tracks. Cart A has a mass of m and cart B has a mass of 2m (twice the mass). The two carts each have a fan that pushes with a constant force (the carts have the same force) and are released from rest. If the forces act for 3 seconds, consider the following.
- How do the final speeds of cart A and B compare?
- How do the final momentums (in the x-direction) for A and B compare?
- How do the final kinetic energies for A and B compare?
Justify your answer.
Now I’m going to tell you the answer (and then we can actually do it).
If cart B has twice the mass, with the same force it will have half the acceleration. This means that at the end of the 3 second time interval, cart A will be going faster. This question shouldn’t be too difficult.
Next — the momentum. Let me start with the momentum principle (in just the x-direction).
Since the two carts have the SAME force and SAME time interval, they will have the SAME change in momentum.
Finally, the kinetic energy. This one is a little tricky — but we can see the answer by using the Work-Energy Principle. This says that the work done on the system is equal to the change in energy of the system. If we pick just the cart as the system then it’s only energy will be kinetic energy (1/2 mv²). The work in 1D will be the force multiplied by the displacement Δr.
So, the cart that moves a GREATER distance (cart A) will have a GREATER change in kinetic energy. But can we also get a numerical comparison for these two values of kinetic energy? Yup, of course. First, remember that the final momentum for the two carts are the same. Second, let’s write the kinetic energy in terms of the momentum.
With the same momentum but different mass, we can get the following ratio of kinetic energies.
So, cart A has the same momentum as cart B but twice the kinetic energy. But can we do it in real life? Yes. Yes we can. Going back to the Vernier carts with fans, they have equal fans that run for a time interval of 3 seconds. One cart has a mass of 500 grams (very close after I added 64 grams) WITH the fan. The other cart has a mass that’s double this (by adding an extra 500 grams).
When they are released, this is what it looks like.
Using the Vernier Sensor Cart, I can also record the velocity for both carts. Here’s a plot.
First, let’s look at the acceleration (the slope of the velocity-time line). The yellow cart has an acceleration of 0.2145 m/s² and the green (with twice the mass) is at 0.1131 m/s². Notice that the lower mass cart should have twice the acceleration — but it’s not quite there. I suspect that the magnitudes of the fan forces aren’t quite the same — I’m going to need to investigate that (in a future blog post). This means that the final speed of the two carts aren’t quite right at 0.63 m/s and 0.328 m/s. Oh well — it’s still close.
I’m pressing on. Nothing can stop me now. Here are the calculations (using the data above — cart A is the yellow cart and B is green).
- PA = 0.107 kg*m/s, PB = 0.1131 kg*m/s (not exactly the same but close if you squint your eyes).
- KA = 0.0115 J, KB = 0.0064 J — at least this works, cart A has more kinetic energy.
Same Force Same Distance
Now for the next question. Suppose two carts have the same force applied over the same DISTANCE? Cart B has twice the mass of cart A. Again, how do the final velocities, final momentum, and final kinetic energies compare?
Cart A will still have a greater acceleration (with its lower mass) — so it seems clear that it will be moving faster. But now we are going the same distance so that the work done on the two carts are the same. This means they should have the same kinetic energy.
In order to think about the momentum, we can look at the time. Which cart will take a longer time to accelerate over the distance (I’m calling it Δr in the sketch above)? With a larger mass, cart B is going to take LONGER to get to the end of the distance. With a larger time, it’s going to have a larger final momentum.
If we want to know a comparison of momentums for A and B, we can again use the kinetic energy in terms of momentum (since they have the same K).
Cool. But is it real? Can we reproduce that? Remember, that’s the whole point of this pointless post. Let’s do it.
Wait. There’s a problem. The Vernier fan is super awesome, however it runs for certain amount of TIME not for a particular distance. Bummer. I guess we are going to have to cheat.
The Vernier fan (technically call the cart fan) has 4 force settings and 4 time interval settings. According to the documentation, these times are for 1 second, 3 seconds, 6 seconds and 60 seconds. Is it possible to use the 3 second time interval and the 6 second time interval to get the two carts with different masses to go the same distance? Yes, that’s where the cheat comes into play.
Suppose I have a cart with an acceleration a1 running for a time interval Δt1. I can use the kinematic equations to get the distance traveled.
We can simplify this since v0 = 0 (starts from rest). Also if we assume the fan exerts a force F and that’s the only force on the cart then the acceleration is a = F/m. If we repeat this for cart 2 and set the distances equal, we get this:
Some stuff cancels and we can set Δt2 = 2Δt1 (twice the time interval).
So there. If I double the time interval and use FOUR times the mass, the two carts should travel the same distance with the same force. I already told you I was going to cheat, so don’t complain. Check it out (oh, I had to pile a bunch of extra masses on cart B to get it to the right mass. Here ar my two carts.
Here is the motion of the two carts (yes, I switched directions).
But did it actually work? Let’s switch over to the data from the sensor carts and check. Here’s a plot of the velocity as a function of time — from this I can get both the acceleration and the final velocities.
With four times the mass and the same force, the green car (B) should have 1/4th the acceleration. I mean, it’s close with a ratio of 0.217 so I guess that’s good enough. The final velocity for the low mass car (A) is 0.649 m/s and cart B is at 0.273 m/s. Using a mass of 0.5 kg and 2 kg, I get the following calculations.
- PA = 0.3245 kg*m/s, PB = 0.546kg*m/s. Again, cart B SHOULD have a greater change in momentum since the fan was on for twice as long. In fact, the momentum should be twice as much — it’s in that ballpark.
- KA = 0.105J, KB = 0.075 J — the kinetic should be the same, this is at least sort of close.
That’s it — well, except to really figure out these fans. Here are my questions:
- What about the frictional forces? Since this depends on the mass of the cart, the more massive cart would have a bigger (proportionally) reduction in the net force.
- What about the force settings? When the fan is on “2 lights” is that twice the force of “1 light”?
- How much variation in force is there between fans? The Vernier documents say there’s also a variation in fan force with respect to battery life — I should check that out.
