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Sunday, 27 September 2015

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 2nd 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.

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