8.The minute hand of a clock is 6 inches long. Starting from noon, how fast is the area of the sector
swept out by the minute hand increasing in in.^2 /min at any instant?POSITION, VELOCITY, AND ACCELERATION
Almost every AP Exam has a question on position, velocity, or acceleration. It’s one of the traditional
areas of physics where calculus comes in handy. Some of these problems require the use of integral
calculus, which we won’t talk about until the second half of this book. So this unit is divided in half;
you’ll see the other half later.
If you have a function that gives you the position of an object (usually called a “particle”) at a specified
time, then the derivative of that function with respect to time is the velocity of the object, and the second
derivative is the acceleration. These are usually represented by the following:
Position: x(t) or sometimes s(t)Velocity: v(t), which is x′(t)Acceleration: a(t), which is x′′(t) or v′(t)Please note that these equations are usually functions of time
(t). Typically, t is greater than zero, but it doesn’t have to be.By the way, speed is the absolute value of velocity.
Example 1: If the position of a particle at a time t is given by the equation x(t) = t^3 −11t^2 + 24t, find the
velocity and the acceleration of the particle at time t = 5.
First, take the derivative of x(t).
x′(t) = 3t^2 − 22t + 24 = v(t)Second, plug in t = 5 to find the velocity at that time.
v(5) = 3(5^2 ) − 22(5) + 24 = −11Third, take the derivative of v(t) to find a(t).
v′(t) = 6t − 22 = a(t)Finally, plug in t = 5 to find the acceleration at that time.
a(5) = 6(5) − 22 = 8