Basic Facts Week 1

Basic facts are those that can appear on a quiz and that must be committed to memory.

Basic Definitions

Distance is a scalar measure of the amount of length separating two points in space.

Displacement is a complete measure of the separation between two points in space. It includes both the magnitude of the separation and its direction. In 1-dimension it is the signed quantity x2 - x1, where x1 and x2 are the position coordinates of the starting and ending points respectively.

Velocity is the rate change of displacement. We usually meet it in two forms.

  1. Average Velocity over some interval of time from t1 to t2. vav=(x2 - x1)/(t2 - t1).
  2. Instantaneous Velocity is the limiting case of average velocity when the interval becomes very small. In calculus terms it is the time derivative of displacement or position.

Acceleration is the rate of change of velocity. In this course we will almost always be concerned with constant acceleration so the distinction between average and instaneous acceleration is rarely interesting. We normally say a=(v2 - v1)/(t2 - t1).

Graphical Representations

We often draw graphs that show position as a function of time with position on the vertical axis and time on the horizontal. In such a graph positive displacements will often correspond to displacements to the right or up in the real world but strictly one should always check.

In a graph of position vs. time the instantaneous velocity is represented by the slope of the line.

If the velocity is constant for some period then the corresponding line segment will be straight. A horizontal segment corresponds to zero velocity.

If the velocity is not constant then the line segment will curve. It is not usually possible to say much more about the acceleration from a position graph.

In a plot of velocity as a function of time the instantaneous acceleration is represented by the slope of the line. Constant accelerations will appear as straight line segments.

Constant Acceleration Kinematics in 1-D

When a particle moves in the absence of acceleration we have, assuming initial position and velocity x0 and v0,

v(t) = v0 = a constant.

x(t) = x0 + v0×t

When a particle moves under constant acceleration a we have

v(t) = v 0 + a×t

x(t) = x0 + v0×t + ½a×t2

Useful Facts

Useful facts are a little more complex than basic facts. It is a good idea to know as many of them as you can but they will not normally appear on quizzes.

In a graph of position as a function of time the curvature of the line is a measure of the acceleration. Thus if the acceleration is positive then the line will be concave upwards and if negative concave downwards.

When a particle moves under constant acceleration a for some displacement (x2 - x1) the initial and final velocities, v1 and v2 are related by

v22 = v12 +2×a×(x2 - x1)