Every object is affected by multiple forces at the same time from different sources and directions, except in free fall in space. The sum of all forces can be combined into a single resultant force. This force results in an acceleration of the object in the same direction as this force points. The acceleration results in a change of the current velocity and hence determines the trajectory of the object.

Here you can lern how the motion of an object can be calculated and how different forces acting on an object can be modeled. I use a simple interactive simulation to help visualize the mechanics of motion. I present some simple experiments and compare the outcome with the simulation. I try to explain the math in layman terms.

Vectors

To compute forces, acceleration, velocity and motion, we have to be familiar with vectors. The basic vector algebra is not much more complicated than ordinary algebra. We only have some more operations and symbols with more than 1 dimension.

If you are not familiar with vectors and want to understand all the equations on this page, I encourage you to read

 What are Vectors?

Vector Notation: To distinguish vectors from scalars (ordinary numbers without multiple dimensions), symbols representing a vector quantity are written in bold fonts on this page.

Definition of a Force

In physics, a force is any interaction that will change the motion of an object. [1]

This includes motions that are only apparent in an non-inertial reference frame. The corresponding apparent forces, like gravity, centrifugal and coriolis forces, are therefore often called fictitious or pseudo forces. In the reference frame they appear, they have to be treated like other forces e.g. contact or friction forces, because they have the same effect: they change the trajectory of objects. Pseudo forces can be measured and even felt, if they are strong enough.

Forces and accelerations are often used interchangeably because they are related to each other according to Newton's second law of motion:

Pseudo Forces and the Role of Reference Frames

An Inertial frame of reference is a reference frame or coordinate system which is not rotating and not accelerated. A reference frame coupled with the surface of the earth is not an inertial reference frame, so we have to include pseudo forces like gravity and centrifugal forces in our calculations.

According to General Relativity a coordinate system on the surface of the earth is an upwards accelerated, non-inertial reference frame [2], because the surface of the earth prevents the system from following the free fall trajectory (Geodesic) of the curved Spacetime produced by the mass of the earth. In an upwards accelerated reference frame objects appear to be accelerated downwards. So there is an apparent downward force called gravity.

A non-rotating, free falling or orbiting reference frame is an inertial frame of reference. That's the reason why in orbit or free fall we don't have a gravitational force, you are weightless.

In a rotating reference frame objects seem to get pulled away from the axis of rotation, while from the perspective of a non-rotating inertial frame of reference they just want to follow a straight trajectory due to Inertia. So in the rotating reference frame there appears a so called Centrifugal force, which has to be taken into account. The centrifugal force is a pseudo force like gravity.

Pseudo forces like gravity and centrifugal forces have to be taken into account like any other forces. They have exactly the same effect on motion, no matter how they are caused.

Gravity of the Earth

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). [3]

We measure and feel the combination (the vector sum) of gravity and the centrifugal force. This combined force is also called effective gravity. The centrifugal force is the reason why the effective gravity depends on latitude. The effective gravity can be measured with a Gravimeter or an Accelerometer.

Equation of Motion

To determine the motion of any object, we have to integrate all forces acting on it. This can be very complicated e.g. to calculate the behavior of an airplane and can only be done by sophisticated computer simulations. But in simple cases, like the one presented here, we can take advantage of the symmetry of the object. We can approximate the object as a point mass and use simple vector addition to calculate the sum of all forces, which than acts on the center of mass.

For simplicity I ignore the rotation of the sphere. Taking rotation into account we can no longer approximate the sphere as a point mass.

Multiple force vectors can be added to a single resultant force vector. The motion of an object is determined by all forces acting on it at the same time.

Terminal Velocity

A falling object is affected by 3 forces: gravity, buoyancy and drag force due to wind resistance.

As long as the gravitational force minus the buoyant force is greater than the drag force, the resultant force is greater than zero and pointing down, so the body is accelerating downwards and it's speed increases. With increasing speed the upwards pointing drag force increases. If the drag force gets equal to the gravitational force minus the buoyant force, the resultant force acting on the body gets zero, which means, there is no acceleration anymore and the speed keeps constant. That is the terminal velocity.

The drag force depends on the speed, the density of the medium the object is traveling through, the shape of the object and its cross section area facing the wind. The shape part of the equation is summarized in the so called drag coefficient $c_\mathrm{D}$. The drag force is:

Or if we know the drag force we can calculate the speed by solving (2) for v:

Lets calculate the terminal velocity of a human falling through air. We assume it is falling streight downward, so all forces lie on the vertical axis. Lets assume the person has a mass of 75 kg. The density of a human body is about the same as of water: 1000 kg/m³.

Lets approximate it's cross section area by a sphere with the same density and mass as the person, so we can use the simulation to cross check the result. The drag coefficient of a human body is greater than that of a sphere, lets assume about 4 times as big: $c_\mathrm{D} = 2$. We can enter this value at Drag Coeff in the simulator Environment panel.

We first need to calculate the cross section area of the spherical body. From the mass and density we can calculate the spherical volume:

From the volume we can calculate the radius of the sphere:

From the radius we can calculate the circular cross section area:

Gravity acts downwards with the force:

The buoyant force of the spherical body always acts against gravity and is:

So we can calculate how strong the drag force has to be to cancel gravity and buoyancy:

We now have all values so we can use equation (3) to solve for the terminal velocity:

The published terminal velocity for a human is about 53 m/s. [4] [5]

The terminal velocity from the simulation is 52.8 m/s.

Kinetic and Potential Energy

We have to distinguish between inside and outside the earth. Outside the gravitational acceleration follows an inverse square law, while inside it is proportional to the distance r from the center:

The potential energy as measured from the surface of the earth to the altitude h ≥ 0 of an object is calculated as follows:

The potential energy as measured from the surface down toward the center (−R ≤ h < 0) is:

This way we have positive potential energy outside the earth and negative potential energy inside the earth while on the surface h = 0 (at radius R) the potential energy is zero.

References