Parachutes and Terminal Velocity

 Some people jump out of perfectly good airplanes on purpose. These parachute jumpers experience a unique sensation during free fall, before their parachutes open. For many parachutists, this is why they jump. During free fall, the parachutist reaches a terminal velocity (a constant velocity) of somewhere between 120 and 180 miles per hour. (If you go feet first, you go faster than if you lie on your back.) When the rip cord is pulled, there is some acceleration, then a gentle float back to earth.

 But what forces act on the parachutist and why is a terminal velocity reached? Gravity and air friction both exert forces on a parachutist. Gravity is constant, but air friction changes with velocity. If you hold your hand outside a car, no air friction acts on your hand if the car is standing still. But if the car is moving at 55 miles per hour, then there is a large force. This shows that this force is dependent on velocity. Actually, air friction is related to the square of the velocity, the size of the object, and a constant. This is shown by the equation:

             f = kAv²

 where f is the frictional force, k is a constant related to the surface over which the air passes, A is the cross-sectional area of the object, and v² is the velocity squared.

 "Cross-sectional area" may be a new idea. Looking up at a parachutist, if she were falling feet first, you would see only a very small area of the sky covered. If she were falling on her back, she would cover a larger part of the sky. We could say that when she is on her back, she presents a larger cross-sectional area. Therefore, from the standpoint of our equation, the frictional force would be greater.

 The frictional force becomes greater as the velocity increases, but the gravitational force remains constant. Acceleration is related to the net force; i.e., the sum of the frictional force and the gravitational force. As the frictional force increases, the net force decreases. As the parachutist goes faster and faster, the net force becomes smaller and smaller, so the acceleration becomes smaller and smaller.

 With a smaller acceleration, there is a smaller change in velocity. The frictional force will not increase much either, so the net force approaches zero, but it will never get there. It will never get to zero for the same reason that a frog will never get out of a well if it keeps jumping half the remaining distance; the closer it gets, the smaller the "change" becomes. And the same is true of the frictional force.

Eventually, the parachutist must pull the rip cord and then things are quite different. What forces does the chute cause? Because the parachute increases the jumper's cross-sectional area, it will provide a larger frictional force at a given velocity (f = kAv²). This force opposes motion, so it is directed upward; it will decrease the downward velocity. Stating this properly, a positive acceleration decreases the negative velocity. Initially, the upward force is very great, so the net force is upward. But as the parachutist slows down, the frictional force decreases until it approximately equals the force of gravity, and a new terminal velocity is reached. The terminal velocity is smaller because there is a larger cross-sectional area. Our parachutist will continue at this constant velocity until she hits the ground, which exerts an upward force on her and stops her.

 Initially the velocity is zero but it becomes more negative  (because it is downward) until a terminal velocity is reached. As terminal velocity is reached, the change in velocity in a given amount of time grows smaller and smaller. After the rip cord is pulled and the parachute opens, the velocity becomes less negative but still remains negative. After a brief moment, the veloc­ity again approaches its terminal point and stays that way until the ground is hit and the velocity becomes zero.                                                                     

The graph of acceleration versus time is also interesting. Initially, the only force acting on our parachutist is gravity, which has a downward pull. This is a negative force and provides negative acceleration, but as the net force becomes smaller, the accelera­tion grows smaller. At terminal velocity, the

acceleration equals zero. After the chute opens, a net upward force causes a positive acceleration which initially is great and then becomes small. A

terminal velocity is again reached and the accel­eration is again zero. Finally, the ground causes alarge upward force and upward acceleration which stops the parachutist. (See Figure 8.2.) As you work with these graphs, think about how the jump would feel. What forces would you expe­rience in a parachute jump? Perhaps you would enjoy the experience, and perhaps you would not?

 

Parachutes and Terminal Velocity – A Review

 I.       Parachutist jumps from plane.

 II.      Person accelerates (gravitational force is count)

     A. Acceleration related to net force

         1. net force = sum of frictional and gravitational forces

         2. as frictional force increases net force decreases

         3. as cross-sectional area increases frictional forces increase

    B. As person goes faster, net force is smaller and acceleration is smaller                                

    C. Net force approaches zero = terminal velocity between 120 & 180 mph

 III.    Rip cord pulled

     A. Chute increases cross-sectional area =larger frictional force

     B. Force directed upward which decreases downward velocity

     C. Net force upward – positive acceleration

 IV.    As parachutist slows down frictional force decreases

     A. Frictional force approximately equals force of gravity 5

      B. New terminal velocity reached

     C. Smaller terminal velocity because of larger cross-sectional area

     D. Person continues at constant velocity till hits ground,

 V.     Ground exerts upward force on person and stops motion

      

Parachutes and Terminal Velocity  --  Questions and Problems

 1. What two forces act on the parachutist during the jump?

 2. How do each of these forces change during the jump?  

3. What happens to the net force while the jumper is in free fall?

 4. What is "terminal velocity" and what causes it?

 5. Why does the downward velocity of a parachutist decrease after the para­chute is opened?

 6. Why does a parachutist reach a second terminal velocity?

 7. Trace what happens to the velocity during the entire jump including: the initial moment in the air, the free fall, the chute opening, falling the rest of the way to the earth, and finally, hitting the earth.

 8. When you ride a bicycle down a hill, you will eventually reach a terminal velocity. If the hill is not very steep, the velocity will be low; a steeper hill will have a higher terminal velocity. Why does this terminal velocity occur?

 9. A feather reaches a terminal velocity very quickly. Explain why this happens.

 10. The volcanic explosion of Mt. St. Helens sent dust up into the stratosphere, which lies more than 7 miles above the surface of the earth. These particles will stay there for a number of years before falling to earth. They fall to the earth because of gravity, but why do they take so long to do so?

 parts of above © 1985 J. Weston Walch, Publisher