An electric car is subjected to acceleration tests along a...

An electric car is subjected to acceleration tests along a straight and level test track. The resulting $v-t$ data are closely modeled over the first 10 seconds by the function $v=24 t-t^{2}+5 \sqrt{t},$ where $t$ is the time in seconds and $v$ is the velocity in feet per second. Determine the displacement $s$ as a function of time over the interval $0 \leq t \leq 10$ sec and specify its value at time $t=10$ sec.

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Key Concepts

Relationship between Velocity and Displacement

In kinematics, displacement is obtained as the integral of the velocity function with respect to time. This fundamental concept shows that knowing how fast an object moves (its velocity) over time allows one to determine how far it has traveled by accumulating (integrating) its speed over the given interval.

Indefinite Integration and Initial Conditions

When integrating a velocity function to find displacement, the integration process yields an arbitrary constant. This constant is determined by applying an initial condition (such as setting the displacement to zero at the starting time). This principle ensures that the computed displacement function accurately reflects the physical scenario.

Integration of Power Functions (Power Rule)

The power rule for integration is used to integrate terms of the form t^n. For example, integrating t raised to a power involves increasing the exponent by one and dividing by the new exponent. This rule is crucial when dealing with polynomial pieces or terms involving t raised to a fractional power, such as when integrating velocity functions that include both integer and non-integer exponents.

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