(A) The blue line has a slope of 5/2, red has slope 4/4 = 1, and purple has slope 3/9 = 1/3.
To find the slope, take any two points (a, b) and (c, d) on the line, then
slope = (d - b)/(c - a)
i.e. the ratio between the changes in the y and x coordinates.
(B) Each line has equation
• blue: y = 5/2 x
• red: y = x
• purple: y = 1/3 x
This follows from the general equation for a line with slope m and y-intercept b,
y = mx + b
or from the more general equation for a line with slope m passing through a point (a, b),
y - b = m (x - a)
(C) Consider the blue line. If you take units into account, its slope is
(5 units of distance)/(2 units of time) = 5/2 (units of distance/time)
which means, per unit time, some object travels a 5/2 units of distance, and this is the definition of speed.
(D) The competitor with the blue line, Mary, would win the race because their speed is the highest (5/2 > 1 > 1/3).
(E) Average velocity is defined as
v (ave) = ∆d/∆t = (change in distance)/(change in time)
At time 0, no one has moved yet. At time 4,
• Mary has covered a distance of 5/2 × 4 = 10, giving her an average velocity of (10 - 0)/(4 - 0) = 5/2
• Mike has covered a distance of 4, so his average velocity is (4 - 0)/(4 - 0) = 1
• Sam has covered a distance of 1/3 × 4 = 4/3, so his average velocity is (4/3 - 0)/(4 - 0) = 1/3
It's not a coincidence that the average velocities are the same as the slopes, because each racer travels at a constant speed.
(F) The racers' instantaneous velocities are the same as their respective average velocities, again because they are each moving at a constant speed.