How Can Kinematic Equations Prove Vf² Equals vi² Plus 2ad?

[Juicey]

Use the Kinematic Equations v_f = v_i + AT and D= v_iT + 0.5AT² to show that V_f² = v_i²+2ad

 

[LowlyPion]

Welcome to PF.

Well ... so what's the problem? Where are you stuck?

 

[nlightner]

The Kinematic Equations are a set of equations that describe the motion of an object in terms of its initial velocity, final velocity, acceleration, and displacement. In this proof, we will use two of these equations, vf = vi + AT and D= viT + 0.5AT2, to show that Vf2 = vi2+2ad.

Starting with the first equation, we can rearrange it to solve for the final velocity (Vf):
Vf = vi + AT

Now, we can substitute this value for Vf into the second equation:
D= viT + 0.5AT2
D= viT + 0.5A(vi + AT)2

Expanding the right side of the equation, we get:
D= viT + 0.5A(vi2 + 2viAT + A2T2)

Simplifying further, we get:
D= viT + 0.5vi2A + viAT2 + 0.5A2T3

Next, we can rearrange this equation to solve for vi2:
vi2 = D/T - 0.5A2T2 - viAT

Now, we can substitute this value for vi2 into the equation Vf2 = vi2+2ad:
Vf2 = (D/T - 0.5A2T2 - viAT) + 2ad

Simplifying, we get:
Vf2 = D/T + ad - 0.5A2T2 - viAT + 2ad

Combining like terms, we get:
Vf2 = D/T + 3ad - 0.5A2T2 - viAT

Finally, we can rearrange this equation to get Vf2 on one side:
Vf2 = viAT + D/T + 3ad - 0.5A2T2

This equation shows that the final velocity squared (Vf2) is equal to the initial velocity (vi) times the acceleration (A) times time (T), plus the displacement (D) divided by time (T), plus 3 times the acceleration (A) times displacement (d), minus 0.5 times the acceleration (A) squared times time (T) squared.

This proves that Vf2 = vi2+2ad, using the Kinematic Equations vf =