The Velocity-Time Graph

In solving problems in linear motion under constant acceleration it is often more advantageous to use a velocity-time graph than to use the equations of motion.
The use of a velocity-time graph is beneficial when there are different parts to the motion.

An example might be the motion of a lift in an office block. It could cover the distance from the basement to the roof by accelerating uniformly from rest to it’s maximum velocity, then travelling at constant speed for the middle section before decelerating, at a constant rate, to rest at the roof garden.

You might like to try a graph like this in your graph copy.
Here is the data required, in the form of a little story.
A lift starts from rest in the basement of a hotel and accelerates uniformly for 15 seconds to reach its maximum speed in 25 seconds. The lift then travels for 30 seconds at this maximum speed for before decelerating uniformly to rest in a time of 10 seconds.

Tan A =  = f

So, the acceleration is the slope of the graph.

Area of rectangle 1 = ut
Area of triangle 2 = ½(v-u).t = ½(ft).t = ½ft2
Area 1 + area 2 = ut + ½ft2 = s
The area under the graph gives the distance travelled.

Two very important points:
Acceleration = Slope of line
Distance = Area under graph

Another way of looking at this is:
The equation of a line in slope-intercept form is y = mx + c
The equation for final velocity is v = u + ft
If we rewrite this as v = ft + u and compare v = ft + u with y = mx + c we can see by inspection that the slope m = f i.e. the slope gives the acceleration.
The area of the graph can be calculated using the area of a trapezium.

Area == s
i.e. Distance = area under the graph.