Acceleration and average velocity

Summary of last lesson

the sign of acceleration
when the acceleration has the same direction of reference system it is positive
when the direction of acceleration is opposite to the reference syste it is negative

Calculating the average of a quantity changing uniformly in time

Geometrical considerations:

click to see animation

The area of the trapezoids doesn’t change, and can be calculated by the area of the rectangle using as height, the average of the bases of trapezoids.

Indeed the equation to find the area of the trapezoid

$A=\frac{m+M}{2}\cdot h$

can be read as the mean base multiplied by the height

This is useful to calculate the average of a quantity that changes uniformly in time;

When a quantity changes in time we should calculate the average in a period of time making the sum of all values assumed during that period and dividing them by their number

Consider this graph that represent the temperature during a period of 6 hours; Here we can’t easly calculate the average temperature during the period, beacuse every temperature is given for different intervals of time.

The situation get worst if the graph is the next one:

But in the next case things are easier

In fact in the last situation the temperature changes uniformly in time, every temperature has a duration of the same time, so seems reasonable to apply the usual equation:

$\bar{T}=\frac{T_1+T_2+T_3}{3}$

So, what can we do here? Well, now the temperature changes continiously, without jumps, but still unifomly: equal time interval correspond in equal change in temperature;
The previous geometrical consideration suggests us to apply the same rule: