Projectile Motion using software, p. 1 Projectile Motion Worksheet ...

Projectile Motion Worksheet General Physics / Adv Physics

Name __________________________________________ Hr __

Doing Projectiles with Projectile Motion Internet Simulation Software (type 1, 2, and 3) Open up the following website: http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html We are going to spend some time learning about this software. Make sure you understand about scaling and how to input different numbers. Make sure you understand how to use the tape measure, etc. We will be doing projectiles of all three types. First work the problem out with your equations and notes and then use the software to verify your numbers. Type 1: This projectile gives you a one-half parabola. For this projectile, the object will have to be launched from a position above the ground and the projectile is launched with a zero degree angle. Let’s try to do one together. Set the angle to be zero degrees and the initial speed at 10 m/s. Lift the cannon so it is 5 meters above the ground with the tape measure. Make sure you understand how to set up the grid correctly. See the following pic: Let’s work out the problem mathematically and then we will see if we get some correct answers.

Now, let’s check the simulation to see if our mathematical result matches up! We will move our target to our mathematically-found range and check to see if our projectile hits the target. Also, let’s check our time to see if our simulation matches up with what we found mathematically. Projectile Motion using software, p. 1

1. How would the time aloft and the range change if we doubled the initial speed? Keep the cannon at the same height of 5 m but double the velocity up to 20 m/s. Make a prediction first about how doubling the initial speed will change the time aloft and the range. Then we will work out the problem mathematically and then we will try the software. Mathematical work:

So, how does doubling the initial velocity change the time aloft and the range?

Now, let’s check the simulation to see if our mathematical results match up! 2. Take the initial speed back down to 10 m/s. Now, let’s double the cannon height to 10 m. How would the time aloft and the range change now? Make a prediction first about how doubling the cannon’s height will change the time aloft and the range. Then we will work out the problem mathematically and then we will try the software.

So, how does doubling the height change the time aloft and the range? Now, let’s check the simulation to see if our mathematical results match up! Projectile Motion using software, p. 2

3. Now why don’t you try one? First work it mathematically to obtain the time aloft and range. Then use the software to see if you can get a score. Angle = 0 degrees Cannon height = 15 m Initial speed = 20 m/s

Time = ____________ sec

Range = _____________ meters

4. Try another one. First work it mathematically to obtain the time aloft and range. Then use the software to see if you can get a score. Angle = 0 degrees Cannon height = 5 m Initial speed = 40 m/s

Time = ____________ sec

Range = _____________ meters

5. I may have to help you with this. In this case, I give you the range and the height. Let’s say our range is 100 m and our cannon height is 10 meters. First, find the time aloft. How fast would the initial speed have to be? Work it out mathematically and then set up the software to check.

Time = ____________ sec

Initial speed = ____________ m/s

6. Let’s say our range is 50 m and our cannon height is 15 meters. First find the time aloft. How fast would the initial speed have to be? Work it out mathematically and then set up the software to check.

Time = ____________ sec

Initial speed = ____________ m/s

7. Here is another twist. Let’s say our range is 25 m and our initial velocity is 25 m/s. How high must the cannon be placed to give the projectile this speed and have it land 25 m away? First find the time. Work it out mathematically and then set up the software to check.

Time = ____________ sec

Height of cannon = ____________ m

Projectile Motion using software, p. 3

Type 2 Projectile Motion. For a type 2 projectile, we have complete symmetry. The projectile is fired from the ground at some angle. It’s trajectory is a complete, full parabola; it starts from the ground and its trajectory ends at the ground. To solve these projectile problems mathematically, we will have to do some different things. Let’s try one together. Let’s say our initial velocity = 10 m/s and our launching angle = 25 degrees. We would like to find the range of the projectile, the maximum height the projectile goes, and the time aloft. Room for our mathematically work here:

Time aloft: _________ Maximum height: _________ Range = ________ Now, let’s check the simulation to see if our mathematical result matches up! Refresh the simulation and put in the correct initial velocity and launching angle. We will move our target to our mathematicallyfound range and check to see if our projectile hits the target. Let’s also check the max height and time. How did the range, max height, and time aloft on the simulation compare with those values found mathematically?

Projectile Motion using software, p. 4

1. How would the time aloft and the range change if we doubled the initial speed? Now, double the velocity up to 20 m/s; keep the launch angle the same (25 degrees). We will work out the problem mathematically and then we will try the software.

Time aloft: ____________ Max Height: ___________ Range: _____________ So, how does doubling the initial velocity change the time aloft, the max height, and the range? Now, let’s check the simulation to see if our mathematical result matches up! 2. Now take the initial speed back down to 10 m/s. In this case, let’s launch it at an angle of 65 degrees. How does that change the max height, the time aloft, and the range that we found on the previous page? Let’s first work the problem out mathematically:

Time aloft: ____________ Max Height: ___________ Range: _____________ Compare the range now to the range on the example, p. 4. How do they compare? ________________This is interesting…Angles that are complementary will give you the same range. What about the time aloft and the max height? How do these change? ____________________________ Now, let’s check the simulation to see if our mathematical result matches up! 3. On your simulation, choose a value for your initial speed (maybe 20 m/s) and choose an angle (θ). Launch your simulated projectile. Write down your range: ____________ time aloft: ___________ and maximum height: ________. Now keep your initial speed the same but change your angle to its complementary angle (90- θ). Launch your simulated projectile. Write down your range: ____________ time aloft: ___________ and maximum height: ________. Complementary angles should give you the same range. However, your time aloft and max height are greater for higher angles. 4. Set up the software to analyze which angle gives you maximum range. Pick an initial speed and don’t change it. By trial and error, find the angle that gives you max range. Don’t erase the trajectories and see if you can find the one angle whereby the projectile goes the farthest. What angle gives you maximum range? __________ Projectile Motion using software, p. 5

Well, why don’t you do several more type 2s for practice without my help. 5. First work it mathematically to obtain the time aloft, height, and range. Then use the software to see if you can get a score. Angle = 30 degrees Initial speed = 40 m/s

Time aloft: ____________ Max Height: ___________ Range: _____________ 6. Try another one. First work it mathematically to obtain the time aloft, height, and range. Then use the software to see if you can get a score. Angle = 50 degrees Initial speed = 60 m/s

Time aloft: ____________ Max Height: ___________ Range: _____________

7. One more. First work it mathematically to obtain the time aloft, height, and range. Then use the software to see if you can get a score. Angle = 80 degrees Initial speed = 40 m/s

Time aloft: ____________ Max Height: ___________ Range: _____________ Projectile Motion using software, p. 6

Time for Type 3 trajectories. Type 3 trajectories involve asymmetric parabolas. Here is an example picture:

To do this with the simulation software, we need to lift the cannon up to some height like we did with type 1 projectiles. But now, instead of a launch angle at 0 degrees, we will have a launch angle of some non-zero number. Here is a picture: Do you see the asymmetry of the trajectory? This is what we find with type 3s.

Projectile Motion using software, p. 7

Let’s do one together and I will show you the mathematically method and then we can check our results using the software. Finding the time is a bit tricky. We’ll have to use the quadratic equation. Cannon height = 10 meters Initial velocity = 20 m/s Launch angle = 30 degrees We would like to find time aloft, and range using mathematics:

Once we get the mathematical method completed, let’s check our results using the software simulation:

Projectile Motion using software, p. 8

1. Let’s try a few type 3s on your own. First work it out mathematically and then check your results using software. Cannon height = 20 meters Initial velocity = 10 m/s Launch angle = 45 degrees We would like to find time aloft, and range using mathematics:

Time aloft: ___________ Range: ____________ Check your work with the software. 2. Here another for practice. First work it out mathematically and then check your results using software. Cannon height = 20 meters Initial velocity = 15 m/s Launch angle = 60 degrees We would like to find time aloft, and range using mathematics:

Time aloft: ___________ Range: ____________ Check your work with the software. Projectile Motion using software, p. 9

3. Here is an interesting one: Let’s shoot the projectile downward: Cannon height = 20 meters Initial velocity = 15 m/s Launch angle = -30 degrees

We would like to find time aloft, and range using mathematics: Time aloft: ___________ Range: ____________ Check your work with the software.

Projectile Motion using software, p. 10

4. Here is a different type of a problem. Now we want the ending height to be higher than the starting height. Cannon height = 20 meters Initial velocity = 15 m/s Launch angle = 60 degrees I want the target to be at a height of 25 meters. Using the quadratic equation, you will find two times. You want the largest time as this is when the projectile is dropping in its trajectory. Find the range where we should put the target.

Find determine it mathematically.

Range: ____________ Use your software and see if you get close! Good job, You know lots about projectiles, now! (Answers to complete worksheet on flip side)

Projectile Motion using software, p. 11

Answers to the worksheet packet

Type 1 projectiles Page 1, example: Time aloft = 1.01 sec; range = 10.1 m Page 2, question 1: Time aloft = 1.01 sec; range = 20.2 m. Same time as before but range is doubled. Page 2, question 2: Time aloft = 1.43 sec; range = 14.28 m. Both the time aloft and the range is increased compared to the example on page 1. Actually, both are greater by √2 Page 3, question 3: Time aloft = 1.75 sec; range = 35 m Page 3, question 4: Time aloft = 1.01 sec; range = 40.4 m Page 3, question 5: Time aloft = 1.43 sec; initial speed = 70 m/s Page 3, question 6: Time aloft = 1.75 sec; initial speed = 28.57 m/s Page 3, question 7: Time aloft = 1 sec; cannon height = 4.9 m

Type 2 projectiles Page 4, example: Range = 7.8 m; Maximum height= 0.91 m; and time aloft = 0.86 sec Page 5, question 1: Range = 31.2 m; Maximum height = 3.64 m; and time aloft = 1.72 sec Page 5, question 2: Range = 7.8 m; Maximum height = 4.2 m; and time aloft = 1.85 sec . The range should be approximately the same as that found on the example, page 4. However, the time aloft and max height will be greater for larger angles. Page 5, question 3: Students can choose any pair of complementary angles. The range should be approximately the same for both. The time aloft and max height should be greater for the larger angle. Page 5, question 4: Given a constant value for the initial velocity, a 45 degree angle should give you maximum range. This is true for type 2 projectiles. Page 6, question 5: time aloft = 4.08 sec; Max height = 20.4 m; and range = 141.3 m Page 6, question 6: time aloft = 9.4 sec; Max height = 108 m; and range = 326.6 m Page 6, question 7: time aloft = 8.04 sec; Max height = 79.2 m; and range = 55.8 m or approx 56 m

Type 3 Projectiles. Use the quadratic equation to find the time aloft. Page 8, example: Using the quadratic gives us two solutions for time aloft: t = -0.73 seconds and t = 2.78 (2.8) seconds. We reject the negative solution. Consequently, the time aloft = 2.78 seconds. To find the range, we multiply by vix and get a range approximately = 48 m (48.5 m) Page 9, question 1: Using the quadratic gives us two solutions for time aloft: t = -1.42 seconds and t = 2.9 seconds. We reject the negative solution. Consequently, the time aloft = 2.9 seconds. To find the range, we multiply by vix and get a range approximately = 20.3 m Page 9, question 2: Using the quadratic gives us two solutions for time aloft: t = -1.09 seconds and t = 3.74 seconds. We reject the negative solution. Consequently, the time aloft = 3.74 seconds. To find the range, we multiply by vix and get a range approximately = 28 m Page 10, question 3: In shooting the projectile downward, you need to find the trig functions using the negative angle. Using the quadratic gives us two solutions for time aloft: t = --2.926 seconds and t = 1.4 seconds. We reject the negative solution. Consequently, the time aloft = 1.4 seconds. To find the range, we multiply by vix and get a range approximately = 18.2 m Page 11, question 4: Using the quadratic gives us two solutions for time aloft: t = 0.47 seconds and t = 2.19 seconds. The smaller time is the motion of the projectile upward and the larger time is the motion as it comes downward. The time we would like is t = 2.19 seconds. To find the range, we multiply by vix and get a target placement of approximately = 16.4 m Projectile Motion using software, p. 12