Modelling Skydiving with Tracker Software

Following my last post explaining the skydiving, here it comes the Skydiving Modelling using TRACKER Software.  

Notes for teachers and learners who would like to use this TRZ file.

 The fy is parameterize for the ease of modelling.

m = mass of human body

A_h = surface area of human body

A_p = surface area of parachute

g at 2000 m height

d = density of air

Cd = drag coefficient of human body

Cd2 = drag coefficient of parachute

g2 = g near surface of Earth

T – duration for parachute opening in Model A

Model A(Red):  Parachute opens slowly over T seconds

Fy function:

if(t<35,m*g-A_h*0.5*d*Cd*vy^2,if(t<35+T,m*g-0.5*d*Cd2*(A_h+(t-35)/T*A_p)*vy^2,if(t<55,m*g-0.5*d*Cd2*(A_h+A_p)*vy^2,if(vy>0,-m*g,0))))

Model B(Blue): Parachute opens fully immediately

Fy function:

if(t<35,m*g-A_h*0.5*d*Cd*vy^2,if(t<55,m*g-0.5*d*Cd2*A_p*vy^2,if(vy>0,-m*g,0)))

Velocity-time graph for Model A: 

Velocity-time graph for Model B:

Comparison of Velocity-time graph for Model A & B:

Acceleration-time graph - Model B compared to Model A    

Model B has a deceleration of 17g, while Model A’s deceleration is 3.1g.

I think it is a bad idea to open the parachute immediate while the parachutist is falling at 40 m/s. In real life, according to a Discovery Education Channel video (https://www.youtube.com/watch?v=ur40O6nQHsw), the parachute takes about 2-3 seconds to open fully. 

Ideas for teaching and demonstration:

Adjust the parameters and compare the velocity-time graph and acceleration-time graph. 

Download the Tracker zip file for Skydiving Modelling.  

You may find the free TRACKER Software at http://www.cabrillo.edu/~dbrown/tracker/

Related Singapore O Level Physics question: 5058/2010/P1 Q3

The Physics of Skydiving Explained with Speed-Time Graph

Singapore GCE O Level Physics /5058 2010 Paper 1 Q 3.

Question: At which point in Fig.1 A, B, C or D is the parachute fully open?

From the moment that the skydiver steps out from the plane, as his weight is the only downward force acting on him, his start to accelerate. As his speed increases, the air resistance increases. The air resistance or drag is proportional to v² and the perpendicular area of a moving object.

D α v².A, where area, A, is constant because he does not use any parachute.

This results in a decreasing downward resultant force. Hence, by Newton’s Second Law, the acceleration is also decreasing as his speed increase. This continues until air resistance reaches the same value of his weight.

At this point the forces are balanced, hence he continues to fall at a constant speed, it reaches terminal velocity (A') in Fig. 2.

From A’ to B, he continues to fall at terminal velocity.

A Point B

At point B, he pulls his parachute.

The air resistance or drag is proportional to v² and the perpendicular area of a moving object.

D α v².A

At B, the gradient of the speed-time graph is negative and constant from the points of speeds of 40 m/s to 25 m/s.

This suggests that the deceleration at point B is constant. Hence, there must be a constant upward resultant force present according to Newton’s 2^nd Law of motion, F=ma.

Immediately after point B, when the parachutist continues to decent, its speed continues to decreases. 

According to the formula for D α v².A, air resistance will decrease when speed decreases. To maintain a constant air resistance, area, A, must increase to compensate the decrease of air resistance due to decrease in speed.

The above means that the area of the parachute is not constant, but increases. This suggests the parachute is in the process of opening from point B. Hence, at point B, the parachute cannot be fully opened.

Point C

At point C, supposed that the parachute is opened fully, the area, A is constant in the drag force equation D α v².A.

The air resistance will decrease as the speed continues to decrease, since area, A, is constant. From point C, the upward air resistance is still greater than the weight of the parachutist.  As the air resistance decreases, the upward resultant force decreases. Hence the deceleration of the parachutist is decreasing, which is evident from the gradient of the speed-time graph from point C, this happens until it reaches another terminal velocity (C’), where the air resistance equals to the weight of the parachutist.     

The above suggests that at point C in Fig. 1, the pareachute is fully open, not point B as the answer provided by Cambridge Marker's Report for the paper.

 He continues to fall at the terminal velocity until he reaches the ground.

For a better visualisation, please watch the video below

Skydiving Modelling using Tracker. The duration of parachute opening is modelled and parameterized in the model.