# Translational Motion Equation Derivation

by helpmeplzzz
Tags: derivation, equation, motion, translational
 P: 42 It's actually a bit of both things you did. Start with this equation $x(t)=x_0+v_0 t+\frac{1}{2}at^2$ Then replace $t=\frac{\Delta v}{a}$, (wich is obvious from the definition of acceleration) so you get $\Delta x = x(t)-x_0=\frac{v_0\Delta v}{a}+\frac{\Delta v^2}{2a}/[itex] Now, just remember that [itex]\Delta v=v_f-v_0$ and multiply by 2a: $2a\Delta x=2v_0v_f-2v_0^2+v_f^2-2v_0v_f+v_0^2$ $\Rightarrow v_f^2=v_0^2+2a\Delta x$ and ther eyou have it :D Hope it helps.
 P: 963 $s=\int_{Vi}^{Vf} \! v \, \mathrm{d} t$ dv/a=dt a is constant. $as=\int_{Vi}^{Vf} \! v \, \mathrm{d} v$ 2as=Vf2-Vi2
 P: 834 All that's needed here is conservation of energy. Starting with a constant force (and hence, acceleration), conservation of energy tells us that initial kinetic energy + work done on the particle = final kinetic energy, or $$E_0 + W = E_f \implies \frac{1}{2} m v_0^2 + m a \Delta x= \frac{1}{2} m v_f^2$$ Cancelling the mass and moving terms around easily gets you the correct result. Most of basic kinematics comes from conservation of momentum ($F = dp/dt$) or conservation of energy, really, and what's taught just corresponds to simple cases of what form the force might take. For instance: 1) No force (F=0): particles move in straight lines 2) Constant force: particles move along parabolas 3) Force proportional to position: springs 4) Central forces: Newtonian gravity, circular motion In the end, whatever the kind of force being studied or what mathematical form it takes, someone somewhere solved the differential equation $F = dp/dt$ for each general kind of force of interest. When combined with the principle of conservation of energy, a scientist has a very powerful set of ideas at their disposal to analyze physical problems. I don't think everyone needs to be able to derive the equations of motion for any arbitrary force, but knowing how to go from these principles to the simple case solutions is useful, and if you face a system with a force like nothing you've seen before, sometimes the only thing to do is to fall back on these basic laws