To determine which description best fits the riders' final velocities, we need to calculate the acceleration of each rider using their initial and final velocities and the time taken. Acceleration can be found using the formula:
[tex]\[ a = \frac{v_f - v_i}{t} \][/tex]
Where:
- [tex]\( a \)[/tex] is the acceleration.
- [tex]\( v_f \)[/tex] is the final velocity.
- [tex]\( v_i \)[/tex] is the initial velocity.
- [tex]\( t \)[/tex] is the time.
Let's calculate the acceleration for each rider step-by-step.
### Gabriella
- Time ([tex]\( t \)[/tex]): 10 seconds
- Initial velocity ([tex]\( v_i \)[/tex]): 55 units
- Final velocity ([tex]\( v_f \)[/tex]): 32 units
Using the formula:
[tex]\[ a_{\text{Gabriella}} = \frac{32 - 55}{10} \][/tex]
[tex]\[ a_{\text{Gabriella}} = \frac{-23}{10} \][/tex]
[tex]\[ a_{\text{Gabriella}} = -2.3 \, \text{units/sec}^2 \][/tex]
### Franklin
- Time ([tex]\( t \)[/tex]): 8.5 seconds
- Initial velocity ([tex]\( v_i \)[/tex]): 50 units
- Final velocity ([tex]\( v_f \)[/tex]): 50 units
Using the formula:
[tex]\[ a_{\text{Franklin}} = \frac{50 - 50}{8.5} \][/tex]
[tex]\[ a_{\text{Franklin}} = \frac{0}{8.5} \][/tex]
[tex]\[ a_{\text{Franklin}} = 0.0 \, \text{units/sec}^2 \][/tex]
### Kendall
- Time ([tex]\( t \)[/tex]): 6 seconds
- Initial velocity ([tex]\( v_i \)[/tex]): 53.2 units
- Final velocity ([tex]\( v_f \)[/tex]): 67 units
Using the formula:
[tex]\[ a_{\text{Kendall}} = \frac{67 - 53.2}{6} \][/tex]
[tex]\[ a_{\text{Kendall}} = \frac{13.8}{6} \][/tex]
[tex]\[ a_{\text{Kendall}} = 2.3 \, \text{units/sec}^2 \][/tex]
### Analysis of Accelerations
- Gabriella's acceleration ([tex]\( a_{\text{Gabriella}} \)[/tex]) is -2.3 units/sec², indicating she is slowing down.
- Franklin's acceleration ([tex]\( a_{\text{Franklin}} \)[/tex]) is 0.0 units/sec², indicating he is not accelerating.
- Kendall's acceleration ([tex]\( a_{\text{Kendall}} \)[/tex]) is 2.3 units/sec², indicating he is speeding up.
### Conclusion
Gabriella is slowing down at the same rate that Kendall is speeding up, and Franklin is not accelerating. Thus, the best description of the riders' final velocities is:
"Gabriella is slowing down at the same rate that Kendall is speeding up, and Franklin is not accelerating."
