Callisto 2 years ago Deriving kinematic equations

1. Callisto

For a = constant

2. Callisto

$v=\int a dt=at+c$c = initial velocity So, $v=u+at$

3. gerryliyana

good job

4. gerryliyana

if a constan, a =0 v = u + at v = u + 0 v = u

5. Callisto

$x_f = \int v(t) dt=\int (\int a dt) dt = \int(at+u)dt = \frac{1}{2}at^2 + ut +c$, c = initial position So, $x_f = ut +\frac{1}{2}at^2 + x_i$$x_f-x_i = ut +\frac{1}{2}at^2$$s = ut+\frac{1}{2}at^2$s=displacement.

6. gerryliyana

for a = 0; s = ut + 0.5 at^2 s = ut + 0 s = ut

7. Callisto

From the third post, $s=ut+\frac{1}{2}at^2$$at^2+2ut -2s = 0$$t^2 + \frac{2u}{a}t - \frac{2s}{a}=0$$(t+\frac{u}{a})^2 - \frac{u^2}{a^2}-\frac{2s}{a}=0$$(t+\frac{u}{a})^2 = \frac{u^2-2as}{a^2}$$t+\frac{u}{a} = \frac{\sqrt{u^2-2as}}{a}$$t = \frac{\sqrt{u^2-2as}-u}{a} -(1)$ Put (1) into v=u+at $v =u+a(\frac{\sqrt{u^2-2as}-u}{a})$$v = u +\sqrt{u^2-2as}-u$$v^2 = u^2 +2as$

8. Callisto

$x_f = \int v(t) dt = v_{ave}t+C$c = initial position $x_f = v_{ave}t +x_i$$x_f = \frac{1}{2}(u+v)t +x_i$$s = \frac{1}{2}(u+v)t$displacement = s = $$x_f - x_i$$

9. Callisto

Probably something wrong with the last post, which is s=(1/2) (u+v)t.

10. experimentX

that's correct ... it assumes constant acceleration. that;s all.

11. Callisto

Seriously?! I did it!?!

12. experimentX

let me see how can i put it up logically.

13. experimentX

|dw:1351833416633:dw|

14. experimentX

now you just need to show that for constant accn V_av = (u+v)/2 ... wanna try it?

15. Callisto

Huh!? I thought it's some maths.. Oh..How to start??

16. experimentX

try using MVT

17. Callisto

MVT.... again.../_\

18. experimentX

well, you can do this without MVT ,,, i was wondering if i could improve my skills with MVT. try expanding v(t) in the expression of average.

19. experimentX

or, V_av = s/t

20. Callisto

expanding v(t)?

21. experimentX

v(t) = u + at

22. experimentX

or simply put ,,, average velocity = distance/time

23. Callisto

Fail ._.

24. Callisto

$v_f= v_i+\int_0^{t_f} a dt=v_i + at$ So, $v_f=v_i + at$

25. Callisto

$x_f = x_i+\int_0^{t_f} v(t) dt= x_i+\int_0^{t_f} (v_i+at) dt = x_i+v_it+\frac{1}{2}at^2$ So, $x_f = x_i+v_it+\frac{1}{2}at^2$ That is $x_f - x_i = v_it+\frac{1}{2}at^2$$s= ut+\frac{1}{2}at^2$

26. experimentX

|dw:1351853269560:dw|

27. experimentX

|dw:1351853406639:dw|