## Velocity

Velocity is often defined as speed with a direction. This is the difference between the two words which are unfortuantely often confused with each other. Velocity is a vector quantity, meaning that it is a quantity that has a numerical value and a direction attached to it.

## Graphing: Velocity vs. Time

The graph for a velocity vs. time graph will almost certainly be linear for any problems encountered in an introductory physics course. The slope of the line indicates the acceleration for the object. Note that a horizontal line has zero slope. Therefore a horizontal line indicates an object that is not accelerating.

In the graph above, a graph of velocity vs. time features a positive slope and a y-intercept of zero. The linear nature of the graph coupled with the positive slope of the line suggests that the acceleration is constant. Furthermore, the fact that the y-intercept is zero indicates that the object started at rest, that is a velocity of zero meters per second.

When an area under the best-fit line is shaded, either a rectangle, triangle, or trapezoid is created. In more advanced examples the shap may be a combination of one or more of these shapes. The area of this shape will be equivalent to the displacement of the object over the enclosed time interval.

Note that in the above diagram, the enclosed shape is a trapezoid. The area of the trapezoid is equivalent to the displacement of the object from 2 to 5 seconds, the boundaries of the trapezoid as indicated by the vertical lines. The bases of trapezoid are equivalent to the initial and final velocity of the interval. From the diagram above, these values are roughly 20 and 50 m/s respectively. The height of the trapezoid is equal to the time interval, that is 3 seconds. Computation of the area of the trapezoid gives 105 m for the displacement of the object.

## Calculating Velocity

Velocity, or more specifically, changes in velocity can potentially be calculated in numerous ways. It is important first and foremost to consider if the situation involves acceleration or not. If the situaton does not involve acceleration, velocity can be found by dividing the total displacement of the object by the total time traveled. For example, a ball that rolls across a 10.0 m horizontal surface in 2.00 seconds has a velocity of 5.00 m/s.

In a situation involving acceleration, finding a velocity is more of a challenge. First of all, by definition in an acceleration scenario the value for velocity is constantly changing. As a result, the change in velocity is usually considered. In order to find such a change, the rate of acceleration must be known. Second, either the time of travel or the distance traveled by the object must be known. If the time traveled by the object is known, then the change in velocity of the object can be found by simply multiplying the acceleration by time. An object that accelerates at a rate of 2.0 m/s2 for 4.0 s experiences a change in velocity of 8.0 m/s. Of the object started at rest, the final velocity will be equal to the change in velocity. Remember that accleration and velocity can be negative, so an acceleration of -2.0 m/s2 over the same 4.0 s interval yields a change in velocity of -8.0 m/s. This means the object slowed down during the time interval. A positive change in velocity means the object moves faster as time progresses.

If the displacement of the object is known, then a change in velocity can also be calculated. For example, the final velocity of an object at rest and allowed to freefall over a distance of 50.0 m can be found. The equation invoked here is v2 = u2 + 2as where v represents the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement. Since the object was released from rest u is equal to zero. Since the object is in freefall, the value for a is equal to g, or 9.81 m/s2. Making the appropriate substitutions, v2 = (2)(9.81)(50.0). When solving for v, the final velocity of the object is found to be 31.3 m/s.