Deceleration A car slows down with an acceleration of...

Deceleration A car slows down with an acceleration of $a(t)=-15 \mathrm{ft} / \mathrm{s}^{2} .$ Assume that $v(0)=60 \mathrm{ft} / \mathrm{s}$ and $s(0)=0.$ a. Determine and graph the position function, for $t \geq 0.$ b. How far does the car travel in the time it takes to come to rest?

Deceleration A car slows down with an acceleration of $a(t)=-15 \mathrm{ft} / \mathrm{s}^{2} .$ Assume that $v(0)=60 \mathrm{ft} / \mathrm{s}$ and $s(0)=0.$
a. Determine and graph the position function, for $t \geq 0.$
b. How far does the car travel in the time it takes to come to rest?

Key Concepts

Deceleration

Deceleration occurs when an object experiences negative acceleration, meaning its speed is decreasing over time. In problems involving deceleration, determining the time at which the object comes to rest is important, and this requires solving for when the velocity function reaches zero.

Acceleration

Acceleration is the rate at which an object's velocity changes with respect to time. It is a central concept in kinematics, and when it is constant, it simplifies the analysis of motion. In calculus, acceleration functions are often integrated to yield the velocity function, thereby linking the concepts of motion directly to integration.

Integration

Integration is the mathematical process used to determine the accumulation of quantities, such as obtaining a velocity function from an acceleration function and a position function from a velocity function. In kinematic problems, integration is used to solve differential equations and incorporate initial conditions to find specific solutions that describe an object’s motion.

Initial Conditions

Initial conditions provide specific values for variables, such as initial velocity and initial position, at a given time. These conditions are essential for determining the constants of integration when solving for velocity and position functions, ensuring the resulting functions accurately reflect the motion of the object from its starting point.

Displacement

Displacement is the net change in position of an object over a period of time. In kinematics problems, the displacement during deceleration is found by evaluating the position function from the start time until the time the object stops. This concept is key in understanding the overall movement of an object in a specified time interval.

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