Assume, we have a motor whose velocity is variable. This motor starts from 0 degree position with 0 degree/Second to 120 degree/Second velocity in 5 second. I want to know how many degrees does this motor move in 5 second? Here is the diagram of velocity and time of motion. enter image description here
Thanks in advance
Let's call the position $\theta$. Then you have the angular velocity $\omega=\frac{d\theta}{dt}$, and the angular acceleration $\alpha=\frac{d\omega}{dt}$. From your graph, you have a constant angular acceleration $$\alpha=\frac{120^\circ/s}{5s}=24^\circ/s^2$$ The angular velocity as a function of time can be obtained from:$$d\omega=\alpha dt$$by integrating between $t=0$ and some time $t$. $$\int_{\omega(t=0)}^{\omega(t)} d\omega=\int_0^t\alpha dt$$ This yields $$\omega(t)-\omega(0)=\alpha(t-0)$$ You have the condition taht at $t=0$ the angular velocity is zero, so $$\omega(t)=\alpha t$$ We repeat now the same reasoning to find the angle as a function of time: $$d\theta=\omega(t)\ dt\\\int_{\theta(t=0)}^{\theta(t)}d\theta=\int_0^t\omega(t)\ dt\\\theta(t)-\theta(t=0)=\int_{t=0}^t\alpha \ t \ dt$$ You have $\theta(t=0)=0$, so $$\theta(t)=\frac{\alpha}2t^2$$