Physics OnlineProjectile Motion
Introduction
The one dimensional constant acceleration kinematics equations can be extended to two dimensions as follows:
x / yvx = vox + axt / vy = voy + ayt
Δx = voxt + ½axt2 / Δy = voyt + ½ayt2
vx2 = vox2 + 2axΔx / vy2 = voy2 + 2ayΔy
Δx = ½(vox + vx)t / Δy = ½(voy + vy)t
Δx = x-component of displacement vector / Δy = y-component of displacement vector
vox = x-component of initial velocity vector / voy = y-component of initial velocity vector
vx = x-component of final velocity vector / vy = y-component of final velocity vector
ax = x-component of acceleration vector / ay = y-component of acceleration vector
t = time interval / t = time interval
These equations are valid when the acceleration vector is constant throughout the time interval. Note that the equations for x and y look similar, with only the subscripts changed. Also note that these general equations are consistent with the specific equations listed in your textbook.
In our experiment, we will study the motion of a projectile. Since the projectile will be in free-fall after launching until impact, we know the acceleration vector will be:
ax= 0
ay = -g = -9.8 m/s2
The value of g will depend on location, so you may need to adjust this value to your local condition.
The manufacturer of the ballistic pendulum claims that the spring gun launches projectiles at specific speeds based on the notch chosen.
Notch 1: vo = 5.40 ± 0.25 m/s
Notch 2: vo = 6.40 ± 0.25 m/s
Notch 3: vo = 7.70 ± 0.30 m/s
If you measure the launch angle and launch height, then the launch distance can be calculated and compared to an experimental value.
Equipment You Procure
- tape measure
- table
- digital camera
- tape
- paper
- pencil or dry erase pen
- textbook
Equipment from Kits
- ballistic pendulum
- bubble level
Experimental Procedures
Zero Launch Angle
1)Place the ballistic pendulum apparatus on a table and move the pendulum and pointer out of the line of fire.
2)Level the ballistic pendulum using the bubble level and leveling screws.
3)Load the spring gun with the metal ball (mass is about 7.7 g) and press it into the first notch.
4)Measure the vertical distance from the floor to the bottom of the metal ball.
5)Use kinematics in the y-direction to calculate the theoretical time to impact. You should already know ay, voy, and Δy. Note that Δy will be negative using a conventional coordinate system.
6)Use kinematics in the x-direction to calculate the theoretical horizontal displacement. You should already know ax, vox, and t.
7)Launch the projectile onto the floor and find the approximate landing spot.
8)Tape a piece of paper to the approximate landing spot.
9)Draw all over the metal ball with a pencil or dry erase pen.
10)Load the spring gun with the metal ball and pull it into the first notch.
11)Launch the projectile. It should land on your piece of paper. If it does not, repeat steps 8 through 11.
12)Measure the experimental horizontal displacement and compare it to the theoretical value.
13)Repeat steps 3 through 12 with the third notch on the spring gun.
General Launch Angle
14) Place the ballistic pendulum apparatus on a table and move the pendulum and pointer out of the line of fire.
15)Raise the far end of the ballistic pendulum by placing your textbook underneath it.
16)Use your tape measure and inverse trigonometry to calculate the angle between the ballistic pendulum and your table.
17)Load the spring gun with a metal ball, but do not press on it yet.
18)Measure the vertical distance from the floor to the bottom of the metal ball.
19)Press on the spring gun until it is loaded at the first notch.
20)Calculate voy = vosinθ and vox = vocosθ.
21)Use kinematics in the y-direction to calculate the theoretical time to impact. You should already know ay, voy, and Δy. Note that Δy will be negative using a conventional coordinate system.
22)Use kinematics in the x-direction to calculate the theoretical horizontal displacement. You should already know ax, vox, and t.
23)Launch the projectile and find the approximate landing spot.
24)Tape a piece of paper to the approximate landing spot.
25)Load the spring gun with a metal ball and press it into the first notch.
26)Use your pencil to cover the metal ball with graphite.
27)Launch the projectile. It should land on your piece of paper. If it does not, repeat steps 24 through 27.
28)Measure the experimental horizontal displacement and compare it to the theoretical value.
29)Repeat steps 14 through 28 with a different notch or launch angle.