After jumping, a skydiver begins gaining which increases the they experience. Eventually they will move fast enough that the air resistance is equal in size to their , but in opposite direction so they have no . This processes is illustrated by for a skydiver with 90 kg mass in the following image:
With a net force of zero the skydiver must be in , but they are not in because they are not static (motionless). Instead they are in , which means that they are moving, but the motion isn’t changing because all the forces are still balanced (net force is zero). This concept is summarized by , which tells us that an object’s motion will not change unless it experiences a net force. Newton’s first law is sometimes called the Law of Inertia because is the name given to an object’s tendency to resist changes in motion. applies to objects that are not moving and to objects that are already moving. Regarding the skydiver, we are applying Newton’s First Law to (back and forth, up and down), but it also holds for the effect of on changes in rotational motion. Changes in motion are known as accelerations and we will learn more about how net forces cause translational accelerations in upcoming chapters.
Using above the statement of Newton’s first law as it applies to net forces and translational motion as an example, write out Newton’s 1st Law as it applies to torques and changes in rotational motion
Dependence of Terminal Velocity on Mass
We already know from our experimental work during the Unit 3 lab that increasing leads to increasing . We can now understand that this behavior occurs because greater mass leads to a greater and thus a greater speed is reached before the () is large enough to balance out the weight.
First-time skydivers are typically attached to an instructor (tandem skydiving). During a tandem skydive the bodies are stacked, so the shape and of the object don’t change much, but the does. As a consequence, the for tandem diving would be high enough to noticeably reduce the fall time and possibly be dangerous. Increasing the to account for the extra mass is accomplished by deploying a small chute that trails behind the skydivers, as seen in the photo below.
A Physical Model for Terminal Velocity
When the skydiver has reached and remains in a state of , we know the size of the must be equal to the skydiver’s , but in the opposite direction. This concept will allow us to determine how the skydiver’s should affect terminal speed. We start be equating the air resistance with the weight:
Then we insert the formulas for and for of an object near Earth’s surface. We designate the in the resulting equation because these two forces are only equal at terminal speed.
We then need to solve the above equation for the .
Let’s estimate the of the human body. We start with the previous equation:
We need to know the , , of air, and of the human body. Let’s use the authors 80 kg mass and the density of air near the Earth’s surface at standard pressure and temperature, . Drag coefficient and cross sectional area depend on body orientation, so let’s assume a standard skydiving posture: flat, horizontal, with arms and legs spread. In this case the drag coefficient will likely be 0.4-1.3. A reasonable value would be . To approximate the cross-sectional area we can use the authors average width of 0.3 m and height of 1.5 m for an area of
Inserting these values into our terminal speed equation we have:
You already have data on how the depends on . We acquired this data using coffee filters in the Unit 3 Lab. Looking back at that data, does that data support our physical model for terminal speed? [Hint: If our (fit equation) suggests that terminal speed depends on mass in the same way as the then yes, our data supports our physical model. Our physical model says that the terminal speed depends on the square root of the mass. Does your empirical fit equation support that result?]
Acceleration During a Skydive
We have now analyzed the skydive after was reached. Prior to this point the forces of drag and weight are not equal, therefore the skydiver is not in and speed will change over time. In order to analyze the early part of the skydive, or the motion of any object that is not in , we need to learn how to quantify changes in and/or direction.
distance traveled per unit time
a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid
the force of gravity on on object, typically in reference to the force of gravity caused by Earth or another celestial body
the total amount of remaining unbalanced force on an object
a graphical illustration used to visualize the forces applied to an object
a state of having no unbalanced forces or torques
the state being in equilibrium (no unbalanced forces or torques) and also having no motion
a state of being in motion, but having no net force, thus the motion is constant
an object's motion will not change unless it experiences a net force
the tenancy of an object to resist changes in motion
motion by which a body shifts from one point in space to another (up-down, back-forth, left-right)
remaining unbalanced torque on an object
a measurement of the amount of matter in an object made by determining its resistance to changes in motion (inertial mass) or the force of gravity applied to it by another known mass from a known distance (gravitational mass). The gravitational mass and an inertial mass appear equal.
the speed at which restive forces such as friction and drag balance driving forces and speed stops increasing, e.g. the gravitational force on a falling object is balanced by air resistance
a force applied by a fluid to any object moving with respect to the fluid, which acts opposite to the relative motion of the object relative to the fluid
The cross-sectional area is the area of a two-dimensional shape that is obtained when a three-dimensional object - such as a cylinder - is sliced perpendicular to some specified axis at a point. For example, the cross-section of a cylinder - when sliced parallel to its base - is a circle
a number characterizing the effect of object shape and orientation on the drag force, usually determined experimentally
relation between the amount of a material and the space it takes up, calculated as mass divided by volume.
mathematical explanation of the relation between measured values that is used for making predictions
mechanistic explanation of how a physical system works