Hot Wheels was one of those all-time classic toys. The original cars were tiny versions of real models like the Chevy Camaro, and you could build tracks to make them do crazy stunts, all driven by gravity. But were those stunts really impossible? It turns out some are indeed possible but maybe not recommended.
Check this out: A human gets in an oversize toy car to go down a track. At the bottom there is a vertical loop to do a full 360. If you were building this in real life (please don’t!), you wouldn’t do it by trial and error the way we used to assemble those orange track segments. You’d model the physics to find out how big the loop should be and how high the car needs to take off.
Let’s see how this would work!
Where to Start
OK, you get a ramp going down to a loop, with the top of the loop maybe 4 meters above the ground. How high up the ramp should you release the car ? Here’s a diagram to help you.
We can think about this in terms of energy. At the top of the track, the car has potential gravitational energy (U). As shown below, this depends on its height (h) and the gravitational force, which is mass (m) x gravitational field (g). As the car rolls down, its height declines, so its potential energy declines.
However, for the Earth-car system as a whole, the total energy remains constant. That’s what we call conservation of energy. So as U declines, the car must gain some other type of energy. That is the kinetic energy (KE) shown above—a quantity that depends on its mass and velocity (v).
When the car reaches the ground, it has zero potential energy and is moving with a certain kinetic energy. Then it goes up the loop and the reverse happens: It slows down and increases in potential energy. For simplicity, let’s consider three key points on this track: Point 1 is at the top of the track, point 2 is at the bottom, and point 3 is the top of the loop. As we just said, the total energy at all three points is the same, so we can write the following:
You can see that the middle point doesn’t matter. Yes, all that potential energy from the starting position goes into kinetic energy at the bottom, but then the kinetic energy just goes back into the same amount of potential energy. So for a loop 4 meters high, it would seem you should start the car at a height of 4 meters.
But that’s actually a terrible idea. Yes, the car will make it to the top of the loop, but because the kinetic energy drops to zero at this point, it won’t go any farther—it’ll just fall straight down.
Keep It Moving
If you want to make it all the way around, the car must still have some velocity at the top. How fast do you need to go? Let’s say the loop is a perfect circle with a radius R. We need to consider the forces on the car at the top of the loop. Here’s a diagram:
There are two forces on the car. There’s the downward gravitational force (mg), and then there’s the downward pushing force from the track (FT). Wait! If there are two forces and both are pushing down, why isn’t the car moving down? Because, remember, forces don’t make an object move—they cause a change in its movement, which we call acceleration.
Indeed the change in the vector velocity of the car is downward. This happens to also be the direction toward the center of the circle. When an object moves in a circle, the direction of the velocity vector changes—which means it’s accelerating, even if it’s moving at a constant speed. We call this centripetal acceleration (ac), and its magnitude depends on both the speed (v) of the object and the radius (R) of the circular motion:
Now, Newton’s second law says that a net force (Fnet) equals mass x acceleration (first equation below), and we can look just at the vertical component of the motion (imagine a y-axis in the picture). Then, plugging in the two downward forces above and our definition of centripetal acceleration, we get:
Both sides of this equation are negative, indicating downward motion. But if you reduce the velocity (v), it’s possible that the centripetal acceleration would be so small that the only way the equation could work is with the force from the track (FT) being in the positive direction. The track would have to pull the car upward. That’s not how Hot Wheels works—the track only pushes away from the surface. This means that if the velocity is too small, the gravitational force will make the car fall instead of moving in a circle.
We can calculate the minimum velocity by setting the track force (FT ) above equal to zero. With this, we can solve for the minimum loop speed as:
Now that we have the minimum velocity, we can plug this back into our energy equation to calculate the initial height of the car. Since the loop is a circle, the height of the loop will be twice the radius (2R). Again, we don’t need to worry about the speed at the lowest point.
I get a minimum height of 2.5R. So if the loop is 4 meters high (with a radius of 2 meters), the car would have to start 5 meters above the ground to just make the loop. Of course, this assumes there’s no energy loss due to friction; you’d probably want to start a bit higher to account for that.
But Not Too High …
In fact, why cut it close? Why not just start much higher and eliminate all doubt? The reason is that the faster the car goes, the higher the g-forces experienced by the driver in the loop.
Let’s think about this: If you release a car so that it goes around the loop at minimum speed, there will be zero force from the track (FT). You’d feel weightless—zero g’s—for an instant. If the car is released from a height greater than 2.5R, its velocity would be greater than the minimum at the top of the loop. In order to still move in a circle, the gravitational force would not be enough. The track would also have to push down on the car. This would create a g-force greater than zero.
Let’s go back to the video of the real stunt. By comparing the loop to the bystanders, I’m guessing it has a radius of 2 meters. The car is clearly released from a height above the 5-meter minimum—let’s say it’s 8 meters. The force at the top of the loop (divided by the weight, to get it in g’s) would be 3 g’s. It’s possible for humans to withstand up to 20 g’s, so this should be fine.
But if you go extreme? If you start too high and make the loop too small, bad things can happen. What about a height of 20 meters with a 1.5-meter radius for the loop? This would produce a force of 21 g’s. It might look cool, but it also might kill you. That’s not fun anymore.