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Velocity is often used interchangeably with the scalar quantity of speed, but the two terms have distinct differences. Speed measures the distance traveled per unit of time and ignores the direction traveled. Velocity, however, is a vector quantity that considers change in position over time (magnitude) and offers a direction of movement. On a straight line without reversing course, speed and velocity are equivalent, but the real world is rarely that neat. Think of a 1-mile circumference race track. When a car crosses the finish line after 500 laps and two hours, it has traveled 500 miles at an average speed of 250 miles per hour. However, because the car ended at its original starting point, the magnitude of its average velocity is zero.

Calculating Straight-Line Velocity

Measure the change in position. On a straight line with a singular direction, this is simply the distance traveled. As an example, if you consistently drove due north from your home for 10 miles, the displacement is 10 miles. If you took a zigzag course to reach the same destination, the distance traveled would be greater, but the displacement would still be 10 miles. Therefore, be careful to measure the straight-line distance between two points when calculating the magnitude of velocity.

Measure the change in time. In the example, if you left home at 2 p.m. and arrived at your destination at 2:30 p.m., it took 30 minutes or 0.5 hours.

Divide the displacement by the change in time to calculate average velocity. In the example, divide 10 miles by 0.5 hours to calculate the average velocity of 20 miles per hour.

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How to Calculate Maximum Velocity

Updated May 22, 2018

By Allison Boley

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

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).

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)=0 when t=pi ÷ 2 and t=(3pi) ÷ 2.

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=pi ÷ 2 and negative just after, so it is a maximum; however, (3pi) ÷ 2 is a minimum because a=3cos(t) is negative just before (3pi) ÷ 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

Enter Velocity Equation

Press the "Y=" button and enter the velocity equation.

Graph Function

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

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.

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 pi. Divide the decimal time by pi. 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 pi 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 pi.

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