PROJECTILE MOTION WITH AIR DRAGA Computer Lab for Calculus Based Physics

Dr. Richard Taylor

Department of Physics & Engineering

Muskingum University

www.muskingum.edu/~richardt

Overview• Developing an Algorithm

• Kinematics• Dynamics• Algorithm

• Implementing the Algorithm• MATLAB• Excel

• Calibrating the Program• Digitizing Real Data• Applying the Program

• Determine the Initial Velocity• Turn Air Drag Off for Comparison

Developing an Algorithm:Kinematics

21 , ,

, 1 , ,

21 , ,

, 1 , ,

1

2

1

2

i i x i x i

x i x i x i

i i y i y i

y i y i y i

x x v t a t

v v a t

y y v t a t

v v a t

1 ,

, 1 , ,

1 ,

, 1 , ,

i i x i

x i x i x i

i i y i

y i y i y i

x x v t

v v a t

y y v t

v v a t

Air drag is a variable force.Therefore, acceleration is not constant!But…If the time interval is small, thenacceleration can be treated as a constant.

Developing an Algorithm:Dynamics

2dragF Bv

1

2B C A

2 2,

2 2,

cos

sin

xdrag x x

ydrag y y

vF Bv Bv Bvv

vv

F Bv Bv Bvvv

0x

y

g

g

F

F mg

,

,

x x

y y

net drag x g xx

net drag y g yy

F F F Bvva

m m mF F F Bvv mg

am m m

Developing an Algorithm:Algorithm

1 ,

,, 1 ,

1 ,

,, 1 ,

2 2, ,

i i x i

x ix i x i

i i y i

y iy i y i

x i y i

x x v t

Bvvv v t

my y v t

Bvv mgv v t

m

v v v

Constants & Initial Values

Implementing the Algorithm:MATLAB

Implementing the Algorithm:Excel

Calibrating the Program

Find the range for the no air drag problem by hand.Find the range using the program.Reduce the time interval in the program until the ranges agree.

0 2 4 6 8 10 12-0.5

0

0.5

1

1.5

2

2.5

3

x (m)

y (m

)

Trajectory

Digitizing Real Data

Applying the Program:Determine the Launch Speed

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x (m)

y (m

)

Trajectory

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x (m)

y (m

)

Trajectory

Adjust the launch speed in the program until the two curves overlap.

Applying the Program:Turn Air Drag Off for Comparison

0 0.5 1 1.5 2 2.5 3-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x (m)

y (m

)Trajectory

Materials can be downloaded fromwww.muskingum.edu/~richardt/aapt2013