For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.
What they knew:
- Linear motion model.
- Vertical motion model.
- Vertical and horizontal motion do not affect one another.
- How vertical and horizontal motion work together to determine the path of a projectile.
- Trig ratios
Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.
It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.
But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.
Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?
This led to a nice conversation on how we need to eliminate as many variables as we can.
Solution: Pick a velocity and work with it. They chose 100 ft/sec.
Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?
So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.
Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.
Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.
Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?
Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.
Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.
They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.
Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)
They were off and running.
- 45 degrees maximizes distance.
- Complementary angles yield the same distance.
- Oh, and this:
I think you physics folks would say something like this: