A car starts from rest and accelerates to a speed of 60 mph in 12 sec . It travels 60 mph for 1

step 1 So for this problem, we're given a piecewise function for a car that is accelerating in miles pe

A car starts from rest and accelerates to a speed of 60 mph...

A car starts from rest and accelerates to a speed of $60 \mathrm{mph}$ in $12 \mathrm{sec} .$ It travels $60 \mathrm{mph}$ for $1 \mathrm{~min}$ and then decelerates for 20 sec until it comes to rest. The speed of the car $s(t)$ (in mph) at a time $t$ (in sec) after the car begins motion can be modeled by: $s(t)=\left\{\begin{array}{cl}\frac{5}{12} t^{2} & \text { for } 0 \leq t \leq 12 \\ 60 & \text { for } 12<t \leq 72 \\ \frac{3}{20}(92-t)^{2} & \text { for } 72<t \leq 92\end{array}\right.$ Determine the speed of the car $6 \mathrm{sec}, 12 \mathrm{sec}, 45 \mathrm{sec}$ and 80 sec after the car begins motion.

A car starts from rest and accelerates to a speed of $60 \mathrm{mph}$ in $12 \mathrm{sec} .$ It travels $60 \mathrm{mph}$ for $1 \mathrm{~min}$ and then decelerates for 20 sec until it comes to rest. The speed of the car $s(t)$ (in mph) at a time $t$ (in sec) after the car begins motion can be modeled by:
$s(t)=\left\{\begin{array}{cl}\frac{5}{12} t^{2} & \text { for } 0 \leq t \leq 12 \\ 60 & \text { for } 12<t \leq 72 \\ \frac{3}{20}(92-t)^{2} & \text { for } 72<t \leq 92\end{array}\right.$
Determine the speed of the car $6 \mathrm{sec}, 12 \mathrm{sec}, 45 \mathrm{sec}$ and 80 sec after the car begins motion.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Kinematics (Acceleration and Deceleration)

Kinematics deals with the motion of objects and often involves phases of acceleration and deceleration. The quadratic expressions in the model represent the acceleration period from rest to a certain speed and the deceleration phase back to rest, illustrating how speed changes over time under constant acceleration or deceleration.

Unit Consistency in Motion Problems

Maintaining consistent units is crucial in motion problems. This involves ensuring that time, speed, and acceleration are all expressed in compatible units, so the calculations and the resulting physical interpretations of the motion remain accurate.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. These functions are useful for modeling situations where behavior changes at certain points, such as when a car accelerates, cruises, and then decelerates.

Function Evaluation

Evaluating a function involves identifying the correct sub-function that applies to a given input value and then substituting that input into the expression. This is essential when working with piecewise functions to determine the correct output for each interval.

Transcript

Please give Ace some feedback