To determine who had a head start in the race and how big the head start was, we need to follow these steps:
1. Calculate the Speeds:
- To find the speed of each runner, we need to determine the change in distance over the change in time.
- John's Speed:
- John's data points are:
- [tex]\( t_1 = 4 \)[/tex] sec, [tex]\( d_1 = 35 \)[/tex] m
- [tex]\( t_4 = 10 \)[/tex] sec, [tex]\( d_4 = 72.5 \)[/tex] m
- The formula for speed is:
[tex]\[
\text{Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}} = \frac{d_4 - d_1}{t_4 - t_1} = \frac{72.5 \, \text{m} - 35 \, \text{m}}{10 \, \text{sec} - 4 \, \text{sec}} = \frac{37.5 \, \text{m}}{6 \, \text{sec}} = 6.25 \, \text{m/sec}
\][/tex]
- Emily's Speed:
- Emily's data points are:
- [tex]\( t_1 = 4 \)[/tex] sec, [tex]\( d_1 = 42 \)[/tex] m
- [tex]\( t_4 = 10 \)[/tex] sec, [tex]\( d_4 = 75 \)[/tex] m
- Using the same formula:
[tex]\[
\text{Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}} = \frac{d_4 - d_1}{t_4 - t_1} = \frac{75 \, \text{m}}{10 \, \text{sec} - 4 \, \text{sec}} = \frac{33 \, \text{m}}{6 \, \text{sec}} = 5.5 \, \text{m/sec}
\][/tex]
2. Determine the Intercepts (Head Start Positions):
- The intercept is where the runner would be at [tex]\( t = 0 \)[/tex] using their speed.
- John's Intercept:
- We use John's initial point and his speed:
[tex]\[
\text{Intercept} = d_1 - (\text{speed} \times t_1) = 35 \, \text{m} - (6.25 \, \text{m/sec} \times 4 \, \text{sec}) = 35 \, \text{m} - 25 \, \text{m} = 10 \, \text{m}
\][/tex]
- Emily's Intercept:
- Similarly for Emily:
[tex]\[
\text{Intercept} = d_1 - (\text{speed} \times t_1) = 42 \, \text{m} - (5.5 \, \text{m/sec} \times 4 \, \text{sec}) = 42 \, \text{m} - 22 \, \text{m} = 20 \, \text{m}
\][/tex]
3. Compare Intercepts to Determine Head Start:
- John’s intercept is 10 meters.
- Emily’s intercept is 20 meters.
- Since Emily's intercept is larger, Emily had the head start.
- The size of the head start is:
[tex]\[
\text{Head Start} = \text{Emily's Intercept} - \text{John's Intercept} = 20 \, \text{m} - 10 \, \text{m} = 10 \, \text{m}
\][/tex]
Conclusion:
Emily had a head start of 10 meters.
