(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 <em>(a, b)</em> and <em>(c, d)</em> on the line, then
slope = (<em>d</em> - <em>b</em>)/(<em>c</em> - <em>a</em>)
i.e. the ratio between the changes in the <em>y</em> and <em>x</em> coordinates.
(B) Each line has equation
• blue: <em>y</em> = 5/2 <em>x</em>
• red: <em>y</em> = <em>x</em>
• purple: <em>y</em> = 1/3 <em>x</em>
This follows from the general equation for a line with slope <em>m</em> and <em>y</em>-intercept <em>b</em>,
<em>y</em> = <em>mx</em> + <em>b</em>
or from the <em>more</em> general equation for a line with slope <em>m</em> passing through a point <em>(a, b)</em>,
<em>y</em> - <em>b</em> = <em>m</em> (<em>x</em> - <em>a</em>)
(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
<em>v</em> (ave) = ∆<em>d</em>/∆<em>t</em> = (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.
