Bike race Theo and Sasha start at the same place on a...

Bike race Theo and Sasha start at the same place on a straight road riding bikes with the following velocities (measured in $\mathrm{mi} / \mathrm{hr}$ ): Theo: $v_{T}(t)=10,$ for $t \geq 0$ Sasha: $v_{S}(t)=15 t,$ for $0 \leq t \leq 1$ and $v_{S}(t)=15,$ for $t>1.$ a. Graph the velocity functions for both riders. b. If the riders ride for 1 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). d. Which rider arrives first at the $10-, 15$ -, and 20 -mile markers of the race? Interpret your answer geometrically using the graphs of part (a). e. Suppose Sasha gives Theo a head start of 0.2 mi and the riders ride for $20 \mathrm{mi}$. Who wins the race? f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

Bike race Theo and Sasha start at the same place on a straight road riding bikes with the following velocities (measured in $\mathrm{mi} / \mathrm{hr}$ ): Theo: $v_{T}(t)=10,$ for $t \geq 0$
Sasha: $v_{S}(t)=15 t,$ for $0 \leq t \leq 1$ and $v_{S}(t)=15,$ for $t>1.$ a. Graph the velocity functions for both riders.
b. If the riders ride for 1 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a).
c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a).
d. Which rider arrives first at the $10-, 15$ -, and 20 -mile markers of the race? Interpret your answer geometrically using the graphs of part (a).
e. Suppose Sasha gives Theo a head start of 0.2 mi and the riders ride for $20 \mathrm{mi}$. Who wins the race?
f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

Key Concepts

Relative Motion and Head Starts

Relative motion examines how the movement of one object compares to another, taking into account any advantages or delays like head starts. Understanding relative motion is crucial for determining outcomes in races or competitions, where timing and initial conditions affect who reaches specific milestones first.

Velocity Functions

Velocity functions describe how speed changes with respect to time. They provide a measure of how quickly an object is moving at any given moment and form the basis for determining the displacement of the object over a period.

Integration for Displacement

Integrating a velocity function over time gives the total displacement or distance traveled. This process involves finding the area under the velocity-time graph, where each slice represents the distance covered in a small time interval.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They allow the modeling of scenarios where the behavior of a quantity, such as velocity, changes at specific times.

Graphical Analysis of Functions

Graphical analysis involves interpreting the shape, area, and trends of functions plotted on a graph. In the context of motion, the area under the velocity-time curve directly represents the distance traveled, enabling geometric comparisons of different motion scenarios.

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