If you’ve been given an equation for velocity to find its maximum (and perhaps the time at which that maximum occurs) calculus skills work in your favor. However, if your math stops at algebra, use a calculator to find the answer. Velocity problems involve anything that moves, from a baseball to a rocket.

Using Calculus

1. Take Derivative of Equation

Take the derivative of the velocity equation with respect to time. This derivative is the equation for acceleration. For example, if the equation for velocity is:

(v=3sin{t})

where t is time, the equation for acceleration is:

(a=3cos{t})

2. Solve Equation for Time

Set the acceleration equation equal to zero and solve for time. More than one solution may exist, which is fine. Remember acceleration is the slope of the velocity equation and the derivative is just the slope of the original line. When the slope is equal to zero, the line is horizontal. This occurs at an extremum, i.e., a maximum or a minimum. In the example

(a=3cos{t}=0text{ when }t=frac{pi}{2}text{ and }t=frac{3pi}{2})

3. Test Solutions

Test each solution to determine whether it is a maximum or a minimum. Choose a point just to the left of the extremum and another point just to the right. If acceleration is negative to the left and positive to the right, the point is a minimum velocity. If acceleration is positive to the left and negative to the right, the point is a maximum velocity. In the example, a=3cos(t) is positive just before t=π /2 and negative just after, so it is a maximum; however, 3π/2 is a minimum because a=3cos(t) is negative just before 3π/2 and positive just after.

If you find more than one maximum, simply plug in times to the original velocity equation to compare the velocities at those extrema. Whichever velocity is larger is the absolute maximum.

Using a Calculator

1. Enter Velocity Equation

Press the “Y=” button and enter the velocity equation.

2. Graph Function

Graph the function. Look at the graph to estimate where the maximum is.

3. Guess Position of Maximum

Press “2nd,” “Calc,” “Max.” Use the arrow buttons to move along the graph just to the left of the maximum and press enter. Arrow just to the right of the maximum, and again press “Enter.” Arrow between those points and enter your best guess of the position of the maximum.

4. Record Values

Record the time (x-value) and velocity (y-value) of the calculator’s more precise solution of the maximum.

If the original velocity equation involves a sine or cosine, watch out for times that the calculator reports involving many decimal places. Your real answer for time may likely involve π. Divide the decimal time by π. If the quotient is close to a fraction, it likely is that fraction, rounded to a decimal by the calculator. Go back to the graph, press “Trace,” and enter the exact fraction — including the π button on your calculator. If you get the same maximum that the calculator found originally, then the maximum does indeed occur at the fractional multiple of π.

Things Needed

• Graphing calculator
• Pencil and paper