Question

Average speed in the time interval t = 0 to t = 40 s ​

Answer 1

Answer:

200m/s

Explanation:

Velocity of a particle is given by the area under the graph till the required time interval as

$v = \int\limits^t_0 {a} \, dt$

However, speed is defined as the magnitude of the velocity (it should always be positive)

Here we can observe that the acceleration is positive for the time interval t=0 to t=20s so the velocity must be positive.

But in the time interval t=20s to t= 40s, the velocity starts decreasing as the acceleration is negative.

Hence the function of velocity is given as,

$v = \left \{ {{20t, t<20s} \atop {400 - 20(t-20),20s<t<40s} \right.$

Average of a function is given as  

${\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\, dx$

Here the average velocity is given by,

$<v> = \frac{\int\limits^{40}_0 {v} \, dt }{40}$

We can split v into two fuctions for the time intervals 0<t<20 and 20<t<40

$\int\limits^{40}_0 {v} \, dt = \int\limits^{20}_0 {20t} \, dt + \int\limits^{40}_{20} [{400-20(t-20)]} \, dt$

which is 8000

now <v> = 8000/40 = 200m/s