Projectile Motion: explaining why Vi=Vf

AI Thread Summary
In projectile motion, the initial velocity (Vi) equals the final velocity (Vf) when the object returns to the same height, as the horizontal velocity (Vx) remains constant while the vertical velocity (Vy) changes due to gravity. At the peak of the trajectory, Vy becomes zero, indicating that half of the flight is complete. The time taken to ascend to maximum height equals the time taken to descend back to the original height, allowing for the use of kinematic equations to demonstrate that Vf equals Vi. The relationship holds true under the assumption that there are no external forces acting on the object other than gravity. This analysis confirms the principle of conservation of energy in projectile motion.
shocklightnin
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How can i show that vi=vf in projectile motion using variables?

Could i just say that since its a parabola we know that vx isn't going to change, but that vy is because it is affected by gravity. so when it reaches max. height, and vy=0, you know that half of the flight is done, and that the object will follow the opposite path it took to get up there, to go back downwards resulting in the same velocity since gravity and vx is constant.

except i don't know how to show this using variables..
 
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shocklightnin said:
How can i show that vi=vf in projectile motion using variables?

Could i just say that since its a parabola we know that vx isn't going to change, but that vy is because it is affected by gravity. so when it reaches max. height, and vy=0, you know that half of the flight is done, and that the object will follow the opposite path it took to get up there, to go back downwards resulting in the same velocity since gravity and vx is constant.

except i don't know how to show this using variables..

you may have to answer why Vx does not change... any forces?

And you could use one of your kinematic equations to show that in the vertical that Vyo is equal to - Vyf when in free fall and assuming the change in position in the vertical is equal to zero.
 
Welcome to PF!

Hi shocklightnin! Welcome to PF! :smile:
shocklightnin said:
How can i show that vi=vf in projectile motion using variables?

(I assume you mean for the initial and final heights being the same?)

One of the standard constant acceleration equations will do it. :wink:
 


You could show that the time needed for the mass to get up to height "h" is the same time it takes for the mass to get from height "h" to the ground. Then using this time calculate Vf, and it will = Vi.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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