Introductory Classical Mechanics/Position, Velocity, Speed and Acceleration

Position edit

The position of any object is the place where we can find it at any constant time.Position is normally defined in terms of co-ordinates of axes.It is a point in space in which an event occurs.Displacement is defined as the shortest distance between two points of positions.It is important to distinguish between displacement, which can be viewed as the distance "as the crow flies" to an object's current location, and distance, which does not take into account direction and is simply what an "odometer" type measurement would read. This key difference is illustrated in the picture below.

Displacement versus distance traveled along a path.

Velocity and speed edit

When dealing with speed (s) and velocity (v), it is necessary to exercise caution. Although both are calculated through similar processes, an object's speed is not always its velocity.

As with acceleration, there are two types of velocity, each with its own definition. Instantaneous velocity (V_inst) is the velocity of an object at a specific time. Average velocity (v_avg) is the change of position (Δx) in relation to the change in time (Δt). The formula for this is V_avg=Δx/Δt. Instantaneous velocity is the limit when Δt approach zero, which is the same as the derivative of position with respect to time.

Velocity and speed measure slightly different things. Velocity is a vector, meaning it has both a magnitude and a direction, while speed is a scalar, with only a magnitude. For example, when talking about the speed of a car, one would be able to simply say that the car is going 50 km/h. However, when using velocity, we must specify the direction that the car is moving. Another important distinction is that, since velocity has a direction, it can "counteract" itself, because it is measured from displacement, not distance (as speed is measured). This is illustrated in the following example: A baseball player is running to a base 30 meters away, at a speed of 10 m/s. They overshoot the base by 10 meters, turn around, and head back to the base, all at an average speed of 10 m/s. However, when we look at the velocity, the person's overall displacement was only 30 meters (as the last 20 meters cancelled out), in 5 total seconds of running (3 for the run, 1 running past the base, and 1 running back to the base), making the average velocity only 30/5 = 6 meters per second, even though the runner's speed was 10 m/s.

Acceleration edit

Acceleration is defined as the change in velocity over the change in time. For example, if a car is at a light that has just turned green, and after 5 seconds, the car was going 15 m/s (around 35 miles per hour), the car's acceleration would be given by Δv/Δt, or 15/5, which works out to be 3. In the SI system of measurement, acceleration is defined in terms of m/s². There are two types of acceleration, average and instantaneous. Mathematically, average acceleration is denoted as a_avg and instantaneous acceleration is a_inst . Unless there is evidence to the contrary, a question about acceleration usually refers to a_avg. Instantaneous acceleration is defined as the derivative of speed (or velocity) with respect to time.

Force and Newton's Laws of Motion edit

An extremely important aspect of acceleration is that it is fundamentally connected with one of the most important aspects of classical mechanics: force. All 3 of Newton's laws of motion reference forces. The first law states:

\sum \mathbf {F} =0\;\Rightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0.

This means that if we treat every force acting on an object as a vector and add all of the vectors together, if the net force is 0, then there will be no acceleration of the object, and the object's velocity will be constant. The reverse is also true: the only way an object's velocity will not be constant is if it is acted upon by a force.

The second law of motion helps us see the relation between force and motion, by stating:

\mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}.

This is more commonly written in the form:

\mathbf {F} =m\,{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=m\mathbf {a} ,

In other words, the force on an object is proportional to the acceleration that that object experiences. In fact, the coefficient of this proportionality is the mass of the object, meaning that force is determined in terms of kg x m/s², which is usually referred to in "Newtons" of force.

Finally, Newton's third law states that for every action, there is an equal and opposite reaction. We are accustomed to this in our everyday lives. If you push on a wall, the wall pushes back at you. Another common example is that, while the explosion of a bullet propels it out the barrel of the gun, it, at the same time, propels the gun back towards the shooter, creating recoil.