Physics
Vandebilt Catholic High School
W. Durpe

Kinematics Notes - Velocity

The branch of physics that deals with objects in motion is called mechanics. Mechanics is composed of kinematics (which describes how objects move) and dynamics (which describes why objects move). In this chapter we study one-dimensional kinematics.

Motion can only be described with respect to some frame of reference. Picture a number line where an object is at position d₁ at time t₁ and at position d₂ at some later time t₂. In SI units we measure position in meters and time in seconds.

Define displacement D d = d₂ – d₁ as the change in position of the object. Note this can be positive or negative, representing motion in some direction. Quantities described by both a number (magnitude) of units and a direction are best represented by mathematical objects called vectors. We indicate vectors by bold type or arrows placed over the variable representing the vector. The SI unit for displacement is the meter.

Displacement is not the same as distance. Distance is a scalar (described by magnitude only). It is never negative. If an object moves from the origin to a position 3 m to the right and then returns to the origin it has traveled a distance of 6 m but its displacement is 0 m. This is because its initial position d₁ = 0 m and its final position d₂ = 0 m. For any round-trip, the displacement is always zero.

Define average velocity v = d₂ - d₁ / t₂ - t₁ = D d/ D t as the rate of change position. This is also a vector, positive or negative indicating direction. It is measured in SI units of m/s. It is not the same as speed. Average speed is defined as distance divided by time and is not a vector. Speed is always non-negative. In the previous example, if the object took 6 seconds to complete the round trip, its average speed would be 6 m / 6 s or 1 m/s. Its average velocity would be 0 m / 6 s or 0 m/s.

To describe the motion of an object more precisely we want to know more than just the average velocity. Did the object always have this velocity, or did it go faster, then slower, pause or change direction? By measuring the position of the object more frequently than just at the beginning and end of its trip we can calculate its velocity over shorter intervals of time.

Define instantaneous velocity "v" as the limit of D d/ D t as D t approaches zero, which is the moment by moment velocity, measured over infinitesimally short intervals of time. Calculus is the branch of mathematics dealing with change. We will not use calculus in this course but calculus students will soon learn that instantaneous velocity as defined here is the derivative of position. For the rest of us, just think of this as average velocity over very short time intervals.

Given a graph of position versus time (meaning position is the vertical axis and time the horizontal axis) our definition of average velocity implies that average velocity over some time interval is the slope of the line (or more generally, curve) between those times. Calculus students will learn about slopes of curves but we will stick with slopes of lines. The slope 'at a point' is the instantaneous velocity. Recall a line that rises to the right has a positive slope, a line that falls to the right a negative slope, a flat line has zero slope, and a vertical line has undefined slope.

Mechanics - the study of why and how objects move

Kinematics - how objects move

Dynamics - why objects move

Vector - has magnitude (a number representing how big it is) and direction

Scalar - has magnitude only

Position - separation between an object and a reference point

Distance - how far you have gone; the magnitude of the change in position; scalar quantity; symbol is d and SI unit is m (for meters)

Displacement - how far you have gone in a certain direction; distance with direction; how far you are from your starting point; vector quantity; symbol is d or x and SI unit is m (for meters); the change in position of an object displacement = change in position = original position minus initial position

Dd = d_f - d_i

Displacement does not always equal the distance traveled! Displacement describes the direction of motion of the object. We will call the displacement of an object moving to the right positive and that of an object moving to the left negative.

Directions in physics can be assigned positive and negative signs. For now, let right and up be positive and left and down be negative.

Speed - distance (how far you have gone) traveled in a given amount of time; scalar quantity; symbol is v and SI unit is m/s (for meters per second)

Velocity - displacement (how far you have gone in a certain direction) per time interval; vector quantity; symbol is v and SI unit is m/s (for meters per second)

In physics, definitions can be expressed mathematically. The definition for speed (and velocity) can be expressed as:

v = d/t

Instantaneous velocity (or speed) - velocity (or speed) at that instant of time, or, it can be defined as the average velocity (or speed) over an infinitesimally short time interval (average speed and velocity are always the same value)

Frame of reference - gives location of an object relative to a reference point. Any measurement of position, distance, or speed must be made with respect to a frame of reference.

Average velocity - total displacement divided by total time. The average velocity of an object is positive if the sign of the displacement is positive and negative if the sign of thedisplacement is negative.

Average speed - total distance divided by total time, or, the distance traveled along a path divided by the time it takes to travel this distance