A car slows down with an acceleration of a(t)=-15 ft / s^2 . Assume v(0)=60 ft / s, s(0)=0, and

step 1 Here we have some facts about the deceleration of a car. First, we're told that its acceleration

A car slows down with an acceleration of a(t)=-15 ft / s^2 ....

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

A car slows down with an acceleration of $a(t)=-15 \mathrm{ft} / \mathrm{s}^{2} .$ Assume $v(0)=60 \mathrm{ft} / \mathrm{s}, s(0)=0,$ and $t$ is
measured in seconds.
a. Determine 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

Displacement Calculation

Calculating displacement involves evaluating the position function over an interval of time. In the context of a decelerating object, the displacement until the object comes to rest is computed by finding the difference in position from the starting time to the time when the velocity becomes zero.

Determining Stopping Time

Determining when a moving object comes to rest involves finding the time at which the velocity function equals zero. This critical value is used to calculate additional parameters such as total distance traveled until the object stops.

Constant Acceleration

In kinematics, constant acceleration is a condition where the rate of change of velocity remains uniform over time. This simplifies calculations because the acceleration does not vary with time, allowing the use of straightforward integration to derive relationships between acceleration, velocity, and displacement.

Integration in Kinematics

Integration is used to derive velocity from acceleration and displacement from velocity. When acceleration is constant or a known function of time, integrating it with respect to time yields the velocity function, and a subsequent integration of the velocity function provides the position function.

Initial Conditions

Initial conditions specify the state of a moving object at a particular moment, such as at time t = 0. They are essential for determining the constants of integration when deriving velocity and displacement functions from acceleration, ensuring that the derived functions satisfy the given starting conditions.

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