b)
To calculate the values of for acceleration, it is helpful to remember what acceleration actually is.
Acceleration is the change in speed over time, with units [tex] \frac{m}{s^2} [/tex]. The y-axis of this graph gives speed, and the x-axis gives time, so we find the point where the car is gaining speed, and we find the rate at which it gained speed.
i) We do this by calculating the slope. We obtain the points where the car starts and stops accelerating, [tex](0,0),(20,40)[/tex], and find the average rate of change (slope) between these two points.
[tex]m= \dfrac{y_2-y_1}{x_2-x_1} = \frac{40}{20} = 2 \ m/s^2[/tex]
ii) We do the same for the deceleration. We choose [tex](40,40),(50,0)[/tex].
[tex]m= \dfrac{y_2-y_1}{x_2-x_1}= \dfrac{-40}{50-40}=-4 \ m/s^2[/tex]
c) We can most easily do this by calculating the distance for each segment of the graph.
For the first 20 seconds, the car moves at an average speed of 20 m/s. [tex]20 \ s \ \frac{20 \ m}{s} = 400 \ m[/tex]
For the middle 20 seconds, the car moves at 40 m/s. [tex]20 \ s \ \frac{40 \ m}{s} = 800 \ m[/tex]
For the final 10 seconds, the car moves at an average speed of 20 m/s. [tex]10 \ s \ \frac{20 \ m}{s} = 200 \ m[/tex]
[tex]400+800+200=1400 \ m[/tex]
You can also do this by finding the area under the graph. You don't need Calculus, because this isn't a curve!
