Projected over two 50 minute class periods
- Materials and Resources
- Begin by throwing a ball through the air to a student. Ask the students what is going on in this situation? Allow students to describe the situation and write down some key points onto the board.
- Ask the students to think back to the vertical motion they studied previously. Throw the ball straight up into the air. Ask the students to describe what is going on in this situation. What are the similarities and differences in the first situation and the second situation? Write down some key points on the board. Point out similarities and differences by circling similarities and visually representing differences as well.
- You have been learning about the Middle Ages in your History, English, Chemistry and other Mathematics courses. Does anyone have an idea where the motion of this ball may connect with something in the battles of the Middle Ages?
- Have computer set up with the following video of a Lord of the Rings scene in which catapults play an integral role: http://www.youtube.com/watch?v=SAW2LxJRLqM (see bottom for media)
3. Lesson Procedure
- Will someone describe their reaction to the movie clip?
- What were your reactions to the catapults that the bad guys used? What were your reactions to the catapults that the good guys used? The catapults that the good guys used are actually called “trebuchets” and constructed a little differently than the catapults. What were the differences? (The main difference is the extension of the rope that creates more initial velocity for the object being launched)
- What were the catapults used for in this video? (To shoot objects at the opposing army and break down their defenses)
- With any weapon, it is important to have a target in mind just like with the arrows and axes from Ms. Rogowicz’s class. Today we are going to explore how to predict where an object launched from a catapult might land.
- (If haven’t already described the motion in the initial questions) What does the path of this ball (throw ball to a student) look like? (Should end up getting to the path being a parabola)
- The path of the object is a parabola. Let’s make a visual representation. Will someone come up and create a visual representation of the path of a ball thrown?
- Student should draw diagram similar to one below.
- What is represented on the x-axis? (Distance)
- What is represented on the y-axis? (Height)
- Has anyone ever been to a place that measures how fast you throw a baseball? If yes, what was your throwing speed? So that was in mi/hr, will practice conversions quickly and convert that to m/s. (For example, if someone said they threw 50 mi/hr the students would need to multiply:
(50 mi/hr) (1 hr / 3600 s) (1609 m / 1 mile) = 22.35 m/s
Let’s pretend our catapult launches a large rock with an initial velocity of 22.35 m/s at an angle of 60°.
Does anyone have a guess as to where the rock might end up? (Write up guesses on board to the side for later comparison)
So what are the things that we know about this rock currently? We know that the initial velocity in this direction is 22.35 m/s, and it is traveling at a 60° angle with respect to the ground. (Label these on the diagram).
- Are there any ideas as to how we might predict where the rock will land?
- Write ideas up on the board, have short discussion on whether or not these are feasible ideas.
- How many directions is this ball moving in? It is moving both vertically and horizontally.
- Can anyone recall what ideas and equations we were using when we studied vertical motion?
- Students should come up with following equation (if they can’t, attempt to re-derive it):
o h(t) = ½ (-9.8 m/s2)t2 + v0 t + x0
- What does each of these variables represent? (-9.8 m/s2 is the gravity constant, t is time, v0 is the initial vertical velocity, x0 is the initial height)
- What constants do we know from the information we have for our example? We know that our initial height is 0 because we’re starting from the ground level.
- Someone may suggest that the initial velocity is 22.35 m/s. Throw the question back to the students: Is the velocity in the y-direction really 22.35 m/s? How might we figure out the vertical velocity? If they are having trouble, draw in the following line to help them see they may be able to use sines and cosines to determine the vertical velocity:
- If they still cannot get it draw in the 90° symbol to signify the triangle in the bottom left is a right triangle. They should then see that the vertical velocity is 22.35 sin 60 = 22.35 (√3 / 2) m/s
- What is the horizontal velocity? 22.35 cos 60 = 22.35 (1/2) m/s.
- Does gravity affect the velocity in the x-direction? Why or why not?
- So going back to our vertical equation, we now know the vertical velocity as well.
- We are trying to find the distance it traveled before hitting the ground. What is the height at the time of landing? (Should come up with 0)
- So what variable is still unknown in our height equation?
o h(t) = ½ (-9.8 m/s2)t2 + v0 t + x0
o h(t) = 0, v0 = 22.35 (√3 / 2) m/s, x0 = 0
- So what variable is still unknown in our height equation? Let’s solve this equation for time.
- In our example, we found out that the rock was in the air for 3.95 seconds.
- What is the next step towards our goal?
- We need to find out the distance traveled in the x-direction in 3.95 seconds.
- What is a formula for distance traveled? D = (r)(t).
- What was the horizontal velocity? How much time was it in the air? Is there anything else we need to take into account (no).
- So the distance traveled is d = (22.35 (1/2)) (3.95) = 44.14.
- What are the units on this distance? (m). How do we know that? We have (22.35/2 m/s)(3.95 s) = 44.14 m (show the seconds canceling).
- So we now know how far the rock would go if we launched it at a 60° angle.
- At what angle do you think the rock would travel the farthest?
- Talk with the person next to you about what angle you think would get you the farthest distance.
- Give the student a minute or two to discuss.
- Now with your partners I would like you to work through the math with the angle you hypothesized to get you the farthest distance.
- Walk around and facilitate questions about the equations and help students work through the math and the steps.
- Meanwhile, create a chart for students to fill in their angles and distances
- When you are finished, put your angle up on the board on the left side of the chart and the corresponding distance you found on the right side.
- Should get a bunch of answers. Have students analyze why these turn out as they do. Hopefully someone guessed a 45° angle and it should be the largest distance.
- Why does this intuitively give us the largest distance?
- Why does this mathematically give us the largest distance?
- Can someone go up to the board to show mathematically how this makes sense? (Should get something like for any initial velocity, the sin45 = cos45 are the two largest numbers you can get with regard to sin θ between 0 and 90 degrees.
- We began our discussion on catapults and hurling an object through the air as a form of weaponry in the medieval times. What might some of the challenges be with this weapon? What are some of the benefits of the weapon? What types of math did the inquiry of the catapult bring to our discussion? How well do you understand the motion of the ball and the corresponding equations? Does it make sense to break up the velocity into horizontal and vertical vectors?