I have been wanting to look at this whole curved bullet thing, but I wasn’t sure how to approach it. In case you are familiar with the myth, this is from the movie WANTED (which I did not see). Apparently, some people learn how to make bullets curve by moving their gun. Here is a shot of a bullet curving in front of someone.
Maybe the picture doesn’t do the clip justice, but it is enough for you to get an idea. Before I do an analysis, this reminds me of a great educational activity. In the activity, you give groups of students a full sheet of paper with lines that look something like this:
You also give the students a ball, like a racquet ball or something. The goal for the students is to make the ball roll through the curve without hitting the lines. This CAN BE DONE. Students then proceed to try a whole bunch of different things, none of them work. It shows the common idea students have that an object can somehow “remember” motion. Give it some curving motion and it will kind of continue with that.
But I said it could happen! That you can get the ball through the curve without hitting the lines. Yes, there is a trick. Maybe you know the answer, but I am not going to give it away. I don’t know where this activity originally came from, but I first saw it done by Bob Beichner as part of the SCALE-UP project.
Ok, back to the bullet. So, why would a bullet curve? For something to change direction of motion, there must be a force acting on it. Instead of calculating the motion of a spinning bullet or something, I am just going to determine the force needed to make a bullet curve. That force can then be compared to other forces like gravity and air resistance.
How does a force make an object curve? If you want to think about it in terms of force and acceleration, the acceleration of an object moving in a circle is:
The negative and the r-hat indicate that the acceleration is towards the center of the circle. So, what do I need to calculate this force to move in a circle?
- The radius of the circle. I will talk about this in a moment.
- The speed of the bullet
- The mass of the bullet
- Maybe the cross sectional area of the bullet and the coefficient of drag if I want to compare this force to air resistance.
For the circle, first I am using a circular path just because it is a little easier. I know the bullet might be able to do something else weird, but I am ok with that. How do I estimate the radius for a bullet? Well, in the WANTED scene, the bullet actually does two things. First it curves away from the person in the middle and then curves back to the target behind her. If I just look at the first curve, I can estimate the distance from the person and the deflection. From this, I can calculate the radius of the circle. Here is a picture.
So, if I estimate d and s I can get the radius of the circle. Here is the location of the two points on a circle that is centered at the origin.
So, the coordinates of the two points would be:
And these would have to satisfy the equation for a circle:
The first point clearly works. Plugging in the second point for x and y:
Now, what about the speed and mass of the bullet? There are so many different types of guns and bullets, I will just pick something. I will just use the info from the bullet in a barrel
post. In that, there was a 9mm fired into a barrel. The round has a mass of 8 grams and a velocity of 358 m/s. If you want to use different numbers, you can enter what you like in the embedded spreadsheet. The other two estimates are for d and s. Let me say d = 5 meters and s = 0.15 meters.
So, from my estimations the force needed to be exerted on the bullet is significantly larger than either the gravitational force or the air resistance force. But, clearly it was fake from the beginning.