11th Phy (24-25) (6-7) PM Offline Class Ch-3 le-5.pdf

### First Equation $V = U + at$ Assume an object moving in a straight line along the *x-axis* with a constant acceleration of *a*. * *U* - initial velocity * *V* - final velocity * *t* - time taken The object covers a distance of *S* in *t* seconds, starting at point *0* and reaching poin...

### First Equation $V = U + at$ Assume an object moving in a straight line along the *x-axis* with a constant acceleration of *a*. * *U* - initial velocity * *V* - final velocity * *t* - time taken The object covers a distance of *S* in *t* seconds, starting at point *0* and reaching point *A*. Based on the definition of acceleration, we have: $a = \frac{dv}{dt}$ $\implies dv = a.dt$ Integrating both sides: $\int_{V_1}^{V_2} dv = \int_{t_1}^{t_2} a. dt$ $\implies [V]_{V_1}^{V_2} = a [t]_{t_1}^{t_2}$ $\implies V_2 - V_1 = a \times (t_2 - t_1)$ $\implies V - U = a \times (t - 0)$ $\implies V - U = at$ $\implies V = U + at$ ### Integration #### Indefinite Integration $\int a.x^n dx = a \times \frac{x^{n+1}}{n+1} + c$ #### Definite Integration $\int_{x_1}^{x_2} a.x^n dx = a \times [\frac{x^{n+1}}{n+1}]_{x_1}^{x_2}$ $= a \times (\frac{ x_2^{n+1}}{n+1} - \frac{x_1^{n+1}}{n+1})$ #### Examples of Integration $\int dx = x$ $\int dt = t$ $\int dv = v$ $\int_{2}^{10} dx = [x]_{2}^{10}$ $= 10 - 2 = 8A$ $\int_{t_1}^{t_2} dt = [t]_{t_1}^{t_2}$ $= (t_2 - t_1)$ $\int_5^6 x^4 dx$ $= 5 \times \frac{x^{4+1}}{4+1}|_5^6$ $= 5 \times \frac{x^5}{5}|_5^6$ $= x^5|_5^6$ $= 6^5 - 5^5$ $\int_2^{10} 2. x dx$ $= 2 \times [\frac{x^{1+1}}{1+1}]_2^{10}$ $= 2 \times [\frac{x^2}{2}]_2^{10}$ $= (10^2 - 2^2)$ $= 100 - 4 = 96$