Three cars, A, B, and C, start from rest and accelerate...

Three cars, $A, B,$ and $C,$ start from rest and accelerate along a line according to the following velocity functions: $$v_{A}(t)=\frac{88 t}{t+1}, \quad v_{B}(t)=\frac{88 t^{2}}{(t+1)^{2}}, \quad \text { and } \quad v_{C}(t)=\frac{88 t^{2}}{t^{2}+1}$$ a. Which car travels farthest on the interval $0 \leq t \leq 1 ?$ b. Which car travels farthest on the interval $0 \leq t \leq 5 ?$ c. Find the position functions for each car assuming each car starts at the origin. d. Which car ultimately gains the lead and remains in front?

Three cars, $A, B,$ and $C,$ start from rest and accelerate along a line according to the following velocity functions:
$$v_{A}(t)=\frac{88 t}{t+1}, \quad v_{B}(t)=\frac{88 t^{2}}{(t+1)^{2}}, \quad \text { and } \quad v_{C}(t)=\frac{88 t^{2}}{t^{2}+1}$$
a. Which car travels farthest on the interval $0 \leq t \leq 1 ?$
b. Which car travels farthest on the interval $0 \leq t \leq 5 ?$
c. Find the position functions for each car assuming each car starts at the origin.
d. Which car ultimately gains the lead and remains in front?

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Key Concepts

Long-Term Behavior and Asymptotic Analysis

Understanding the long-term behavior of a moving object requires evaluating the limits of its velocity and position functions as time increases. Asymptotic analysis helps identify which object may ultimately dominate in motion by focusing on the behavior of the functions once transient effects have diminished.

Rational Functions in Velocity Modeling

Velocity functions expressed as rational functions, which are ratios of polynomials, capture complex acceleration behaviors. Analyzing these functions involves understanding their asymptotic properties, limits, and how the numerator and denominator interact, which are crucial for predicting motion characteristics.

Relationship between Velocity and Position

In kinematics, an object’s position is described as the integral of its velocity over time. This fundamental relationship allows us to determine the displacement or distance traveled by an object by accumulating its instantaneous velocity over a specified time period.

Definite Integration to Determine Distance Traveled

Definite integration is used to compute the total displacement over an interval by calculating the area under the velocity-time curve. This method is essential when comparing motion over fixed intervals, as it quantifies the overall distance covered given varying rates of change.

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