In AP Physics C, we use calculus to describe motion. Displacement, $\Delta x$, is the change in position. Velocity, $v = \frac{dx}{dt}$, is the rate of change of displacement with respect to time. Acceleration, $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$, is the rate of change of velocity with respect to time. We can also integrate acceleration to find velocity and velocity to find displacement, remembering to consider initial conditions.
Match the terms with their definitions:
| Terms | Definitions |
|---|---|
| 1. Velocity | A. Rate of change of velocity with respect to time |
| 2. Displacement | B. Change in position |
| 3. Acceleration | C. Rate of change of displacement with respect to time |
| 4. Integral | D. The area under a curve; used to find displacement from velocity or velocity from acceleration. |
| 5. Derivative | E. The slope of a curve; used to find velocity from displacement or acceleration from velocity. |
Match the correct definitions to the terms: 1-C, 2-B, 3-A, 4-D, 5-E
Complete the paragraph below using the following words: acceleration, velocity, displacement, calculus, derivative, integral.
Understanding motion in Physics C requires the use of ________. The rate of change of ________ is called velocity, which can be expressed using a ________. The rate of change of velocity is called ________. To find the velocity from acceleration, we use an ________.
Answer: calculus, displacement, derivative, acceleration, integral.
Imagine a car moving along a straight road. The car's acceleration is given by $a(t) = 2t - 1$, where $t$ is time in seconds and $a(t)$ is in $m/s^2$. If the car starts from rest (i.e., $v(0) = 0$) at position $x(0) = 0$, determine the car's velocity and position as a function of time. Explain each step.
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