Name:

Section:

ID:

Date:

Projectile Motion

Objectives

· To investigate the relationship between the angle of the projectile and its range, and calculate the value of acceleration due to gravity g.

· To explore vector representations in two dimensions.

· To study the effect of air resistance on projectile motion.

Introduction

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. The object is called a projectile, and its path is called trajectory. We neglect the effect of air resistance on the projectile motion and using the equations of motion, we can derive equations for the optimum range R and height H for the projectile.

(1)

(2)

(3)

If the object fired with angle θ, its initial velocity can be described in term of cos θ and sin θ Figure 1:

(4)

(5)

Figure 1

By analyzing the motion along x and y axes, and using the following trigonometric identity

sin (2)=2 sin cos

one can easily derive the equations for the projectile range R and height H

(6)

(7)

PhET Virtual Lab simulation:

We will use PhET interactive simulations for the projectile motion experiment. Access the simulations by this link:

https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html

or download the offline version from our physics 101 web site:

http://www.kfupm.edu.sa/sites/phys101/SitePages/en/ContentDetailsPage.aspx?CUSTOMID=26&LinkID=LinkV7

A click on the play button will open a new window where you will see four different sections of the lab. Run all simulations one by one and take a few minutes to check out the capabilities of the simulator. Specifically, explore the measuring tools for the trajectory as shown in Figure 2. Learn how to use them with the pause/play buttons. Using these tools, you can take data at any point along the trajectory.

Figure 2

Intro Screen: Figure 3

Investigate the factors that affect trajectory of a projectile, such as angle, height, initial speed, and air resistance.

Figure 3

Lab Screen: Figure 4

Explore the effects of different parameters on the projectile motion, and investigate the influence of gravity.

Figure 4

Exercise 1: the relationship between the angle of the projectile and its range (Optimal angle)

In this Exercise, students will investigate how the range of the projectile is affected by its initial launching angle. Given the initial speed, students will determine the angle at which the range of the projectile is maximum and conclude the value of acceleration due to gravity g. The students will also measure the maximum height of the projectile in the experiment and will compare it with the theoretical value calculated by using equation 7.

On the home page of Projectile motion simulations (PhET link), a click on the “Lab” screen will open a new window as shown in Figure 5. First, select cannonball as your projectile from the right side menu, then adjust your simulation parameters using the following guidelines:

1- Make the height of the cannon h=0, and the magnitude of the initial velocity of the lunched object equal to (last digit in your ID number+10) m/s.

2- Adjust the first angle to 25o, and fire your projectile

3- Record the maximum range and maximum height of the projectile for this angle in table 1.

4- Run the simulation, measure the projectile range (R) - Figure 5, record the angle and the range with 2 decimal places in table 1.

5- Do not erase your data. At the end of your experiment, you will copy and paste the data screen in your lab report.

6- Change the angle as shown in table 1, and repeat step 4 for all angles listed in the table.

7- Calculate sin(2θ) and sin(θ). You can do these calculations by Excel.

Figure 5

Table 1

Vi = (last digit in your ID number+10) m/s

Vi = (m/s)

θ

Sin (2θ)

R (m)

Sin (θ)

Hmax (m)

25

30

35

40

45

50

55

60

65

70

75

80

85

90

8- Plot R versus sin(2θ) graph by using Excel.

9- Apply linear line fit to the data and determine its slope.

10- Use the slope and equation (6) to answer the following questions.

Paste your graph here

Paste your data acquisition screen below.

Q.1: What is the slope of your graph and what does it represent?

Slope= , it represents

Q.2: What is the experimental value of g (from your simulation)?

=

Q.3: What is the percentage error in the measured quantity g if you consider = 9.80 m/s2

(8)

% error in g =

Q.4: What is the maximum range R, and what angle does correspond to the maximum range? Explain why this particular angle produces the maximum range.

R=

=

because

Q.5: What is the value of the maximum height (Hmax), and what angle does give you the maximum height?

Hmax

From simulation=

By using formula (show steps)=

=

Q.6: When a projectile covers maximum range, what is the maximum height of its trajectory?

Hint: Not sin

Hat Rmax

From simulation =

By formula (show steps)=

Q.7: What is the flying time for the maximum range (R)

tRmax=?

From simulation=

By formula (show steps) =

Q.8: Two balls are dropped from the same height and at the same time one vertically down and the other one horizontally (see Figure 6), which ball will hit the ground first? Explain.

Figure 6

Exercise 2: calculate the horizontal component of the velocity (Vx) of a projectile throughout its flight

In this Exercise, students will measure the x- component of the velocity for the projectile by using the equations of motion and solve for vx in (m/s) as a function of R in (m) and t in (s):

(9)

(10)

Figure 7

On the home page of Projectile motion simulations (PhET link), a click on the “Intro” screen will open a new window as shown in Figure 7. On the intro screen, select cannonball as your object, use the provided controllers to adjust the height (h = 0 m) and the angle of the projectile at 40o and the initial velocity to be 15 m/s.

1- Run the simulation and get the projectile trajectory.

2- Use the “time, range and height” measuring tool to measure the time and the x-position (range at that time instant) of the projectile every 0.2 sec. Starting from the initial point throughout its flight. Fill your data in the table 2.

Table 2

θ = 40o , = 15m/s

Time (s)

X(m)

(m/s)=X/t

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Q.9: From the x-component of the velocity, what do you conclude?

Exercise 3: Explore vectors representations

From the home page of PhET link, click on vectors screen, select cannonball as your object, use the provided controllers to adjust the height (h=0m) and the angle of the projectile at 50o and the initial velocity to be 20m/s. Set the vector display options to show components, velocity vectors and acceleration vectors, turn off force vectors and air friction (see figure 8).

Figure 8

Fire the cannonball (at an initial speed of 20 m/s) and record your observations of the vectors. Refire the cannon at this position as many times as you need to make your observations. Maybe try using the slow motion setting. Try using the pause/play button too.

Q.10: How does the acceleration vector change throughout the cannonball’s trajectory?

Q.11: How do the velocity vectors change throughout the cannonball’s trajectory?

Q.12: Describe horizontal velocity vectors (vx) then Draw the vx vectors at every two dot along the trajectory?

Q.13: Describe vertical velocity vectors (vy) then draw the vy vectors at every dot along the trajectory. Include vector at the top?

Exercise 4: Effect of air resistance (optional)

From the home page of PhET link, click on Intro screen, select Piano as your object (object with big area). Use the provided controllers to adjust the height (h = 0m) and the angle of the projectile at 60o and the initial velocity to be 18 m/s. Run the simulation once with checking the air resistance box (air resistance will be present), and another time run it without the air resistance box unchecked (Off) (ignore the air resistance). Use the measuring tool to find the maximum height (H) and the maximum range (R) of the projectile (see figure 9) and fill the required information in table 3.

Figure 9

Table 3

Range R(m)

Maximum height H(m)

Piano without air resistance

Piano with air resistance

Q.14: What can you conclude about the effect of air resistance on the projectile trajectory?

Conclusion: What did you learn from this experiment?

© KFUPM –PHYSICS

Department of Physics

KFUPM, Dhahran 31261

written by Ayman Ghannam and

Dr. K. Alam on 11/10/2020

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