We want to know who will reach the finish line first. This can be shown graphically below.
The y-axis represents the distance and the x-axis represents time. You can see that Usain Bolt starts a little later than the senior but has a steeper slope (faster speed) and catches up.
The point of intersection of these lines is when the distances are the same. After this point, Usain will win. It's a close one at \(100\;[m]\)!
Let's set up a table to organize the information we know. Let \(d\) represent the distance Usain bolt passes the senior. Let \(t\) represent the time the senior has ran. Let \(v\) represent the senior's speed.
| Athlete |
\(d\) |
\(v\) |
\(t\) |
| Usain Bolt |
\(d\) |
\(10.4\) |
\(t - 1\) |
| Highschool Super Star |
\(d\) |
\(v\) |
\(t\) |
Notice that we could have let \(t\) represent the time Usain Bolt has been running. Then, the time for the senior would be \(t + 1\).
There are a few unknowns: \(d\) is the distance ran (for the point of intersection, it is the distance at which Usain Bolt passes the senior) \(t\) is the time (after the senior started running) he passes him and \(v\) is the senior's speed.
We can use information in the question to solve for the senior's speed:
\( \frac{d}{t} = v \)
\( \frac{100 \; [m]}{10.4 \; [s]} = v \)
\( v = 9.6 \; [\frac{m}{s}]\)
Update the table:
| Athlete |
\(d\) |
\(v\) |
\(t\) |
| Usain Bolt |
\(d\) |
\(10.4\) |
\(t - 1\) |
| Highschool Super Star |
\(d\) |
\(9.6\) |
\(t\) |
Let's set up the equations:
| Usain Bolt |
\(d = 10.4 \cdot (t-1) \) |
| Highschool Super Star |
\(d = 9.6 \cdot t\) |
Two equations with two unknowns means we can use substitution or elimination. Since the equations are both already in terms of \(d\), substitution is a good pick.
\(d = 10.4 \cdot (t-1) \)
\(9.6 \cdot t = 10.4 \cdot (t-1) \)
\(9.6 \cdot t = 10.4 \cdot t - 10.4 \)
\(9.6 \cdot t - 10.4 \cdot t = - 10.4 \)
\(t = \frac{- 10.4}{-0.8} \)
\(t = 13 \; [s]\)
This represents the time at which Usain Bolt passes the senior. Plug back into either equation to find the distance.
\(d = 9.6 \cdot t\)
\(d = 9.6 \cdot 13\)
\(d = 124 .8 \; [m]\)
Since Usain Bolt won't pass the senior until \(124\;[m]\), the senior will win the \(100\;[m]\) race.