Example 4

A particle moves along the y-axis with constant velocity. At time t = 0 sec, it is at the point y = 3 ft. At time t = 2 sec, it is at the point y = 11 ft. Find the velocity and the equation for the motion.

The velocity is defined as the distance moved divided by the time elapsed, so the velocity is

If the motion of the particle is plotted in the (t, y) plane as in Figure 1.3.7,

the result is a line through the points P(0, 3) and Q(2, 11). The velocity, being the ratio of Δy to Δt, is just the slope of this line. The line has the equation

y - 3 = 4t.

Suppose a particle moving with constant velocity is at the point y = y₁ at time t = t₁ and at the point y = y₂ at time t = t₂. Then the velocity is v = Δy/Δt. The motion of the particle plotted on the (t, y) plane is the line passing through the two points (t₁, y₁) and (t₂, y₂), and the velocity is the slope of this line.

An equation of the form

Ax + By + C = 0

where A and B are not both zero is called a linear equation. The reason for this name is explained by the next theorem.