Acceleration

We cannot even approximation real motion by intervals of constant velocity. No real object can instantaneously change from one velocity to another. The change in velocity takes place during an interval of time. For example in the interval of time from t₁to t₂, delta-t, the velocity may change from v₁ to v₂, delta-v. The ratio delta-v/delta-t is called the acceleration. If the acceleration is not constant and the time interval is finite then this ratio is the average acceleration. Acceleration is the slope of the velocity-time graph. Recall that velocity was the slope of the position-time graph. If acceleration is not constant then the instantaneous acceleration at any time is the slope of the tangent line on the velocity-time graph at that time.

For example, the Power Wheels started from a dead stop and reached a velocity of 1 m/s in 2 s. The average acceleration is (1 m/s)/(2 s) = 0.5 m/s/s. The car reached a higher value of instantaneous acceleration during the 2 s interval which you might try to estimate by drawing a tangent line. We usually write 0.5 m/s². Writing the units of acceleration as m/s/s is acceptable and is not ambiguous if one applies the usual rule of evaluating operators from left to right.

So far the concept of acceleration is probably pretty familiar to you because of its common usage to describe speeding up. The term is used in physics in a much broader sense. For example if v₁ is larger than v₂ then we would have a negative acceleration. This happens, for example, when one is travelling in the positive direction and is slowing down. Colloquially, we describe this with the term deceleration, but technically, the term acceleration applies to this case as well. Furthermore, travelling in the negative direction and slowing down gives a positive acceleration and travelling in the negative direction and speeding up is negative acceleration. The sign of the acceleration is not necessarily the same as the velocity's. This means that the acceleration's direction is not always the same as the direction of the veolcity. When we start talking about two and three dimensional motion this fact becomes even more important. To reiterate, in common usage, acceleration means going faster in the forward direction. In physics it means the ratio of velocity change and time interval during which that velocity change occurred. That's not always the same thing.

Relations for constant acceleration

The graphical way of deriving displacements from a velocity-time graph is completely general. No matter what the motion is, as long as it can be represented graphically, we can estimate the displacement during any time interval by estimating the area under the curve.

Formulas for constant acceleration can be derived and are useful even though constant acceleration rarely occurs exactly. Falling objects, if they are heavy and dense enough, may approach constant acceleration fairly closely. In other cases, such as the braking of a train, assumming constant acceleration may be a useful first approximation to the actual braking motion. Thus the formulae for constant acceleration are usually a staple of first year physics courses. (They also form a convenient topic for problems early on in the course.)

First sketch the velocity-time graph for constant positive acceleration and positive velocity. The displacement between any two times is got by figuring out the area under the curve. This area is a quadrilateral which is not rectangular unless the acceleration is zero. We can figure out the area by first getting the area of the rectangle and adding the area of the triangle. (Remember that the area of the triangle is 1/2 base x height.)

Area of quadrilateral = Area of rectangle + Area of triangle
delta-x = (delta-t) (v1) + (1/2)(delta-t) (delta-v)
recall that which comes directly from the definition of acceleration.
Thus
delta-x = (v1)(delta-t) + (1/2)a(delta-t)2
This is the equation of a parabola, and if you look at the graph of position vs time for constant acceleration you will see that a parabola is a reasonable curve to fit the graph.
Even though we used a picure of positive acceleration and positive velocity to derive this equation, you should verify that it is general and applies in cases of negative acceleration or negative velocity or both.

Problem: Draw velocity-time graph for negative acceleration, positive velocity and verify that the equation applies. Repeat for the other two combinations of signs of acceleration and velocity.

This equation is so important and popular among physics teachers that it is usally called one of the "Kinematic equations". Sometimes slightly different notation may be used. For example, the one I remember goes like this:
x = x0 + v0 t + (1/2)a t2

x = x<sub>0</sub> + v<sub>0</sub> t + (1/2)a t<sup>2</sup>

In this case t is used for delta-t and delta-x =x - x₀.
Remember that this equation applies only for constant acceleration.

Another kinetmatic equation applies for constant velocity, zero acceleration is

delta-x = v (delta-t)

As I said, many traditional early problems in your physics courses involve manipulating these "kinematic equations" and you may wish to memorize them. If you forget the equation relating displacement to constant acceleration, just redraw the little graph and figure out the areas as we did above.