Word Problems - Motion 1

In this question, we are asked for the speed, \(v\). We are given \(t = 9.58 \; [\text{s}]\) and \(d = 100 \; [\text{m}]\).

First, we can use the distance formula to isolate for \(v\):

\( d = v \cdot t \)

\(v = \cfrac{d}{t}\)

Next, we can plug in the known values to solve for \(v\):

\(v = \cfrac{100 \; [\text{m}]}{9.58 \; [s]}\)

\(v = 10.4 \; \left[\cfrac{\text{m}}{\text{s}}\right]\)

Therefore, we can determine that Usain Bolt was running at an average speed of \(\boldsymbol{10.4 \left[\cfrac{\textbf{m}}{\textbf{s}}\right]}\).

We want to know who will reach the finish line first. This can be shown in the graph below:

From analyzing the graph, we can determine the \(y\)-axis represents 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 \(100\;[\text{m}]\)!

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:

\(v = \cfrac{d}{t}\)

\(v = \cfrac{100 \; [\text{m}]}{10.4 \; [\text{s}]}\)

\( v = 9.6 \; \left[\cfrac{\text{m}}{\text{s}}\right]\)

Next, we can update our table of values as such:

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	\(d = 9.6 \cdot t\)

Since both equations each contain an unknown variable, we can use either substitution or elimination to solve for it. Since the equations are both expressed in terms of \(d\), we will use substitution:

\(d = 10.4 \cdot (t-1) \)

\(9.6 \cdot t = 10.4 \cdot (t-1) \)

\(9.6 \cdot t = 10.4 \cdot t - 10.4 \)

\(9.6 \cdot t - 10.4 \cdot t = - 10.4 \)

\(t = \cfrac{- 10.4}{-0.8} \)

\(t = 13 \; [\text{s}]\)

This represents the time at which Usain Bolt passes the senior. Plug back into either equation to find the distance.

\(d = 9.6 \cdot t\)

\(d = 9.6 \cdot 13\)

\(d = 124 .8 \; [\text{m}]\)

Since Usain Bolt won't pass the senior until \(124\;[\text{m}]\), the senior will win the \(100\;[\text{m}]\) race.