Which equation relates final velocity, initial velocity, acceleration, and displacement?

Which equation relates final velocity, initial velocity, acceleration, and displacement?

• A. vf = v0 + a t
• B. vf^2 = v0^2 + 2 a delta s
• C. vf = v0 - a t
• D. vf = v0 + a delta s

Answer: (B)

You're testing how final velocity relates to initial velocity, acceleration, and how far you’ve moved, without involving time. For constant acceleration, the relationship vf^2 = v0^2 + 2 a Δs directly ties those quantities together. It comes from the chain rule a = dv/dt = (dv/ds)(ds/dt) = v dv/ds, and integrating v dv = a ds from v0 to vf and s0 to s0+Δs. This yields (1/2)(vf^2 − v0^2) = a Δs, which rearranges to vf^2 = v0^2 + 2 a Δs. The other forms use time (t) or mix in Δs in a way that doesn’t give a correct velocity-squared relation, and one would be dimensionally inconsistent if you tried to treat a Δs as a velocity.

You're testing how final velocity relates to initial velocity, acceleration, and how far you’ve moved, without involving time. For constant acceleration, the relationship vf^2 = v0^2 + 2 a Δs directly ties those quantities together. It comes from the chain rule a = dv/dt = (dv/ds)(ds/dt) = v dv/ds, and integrating v dv = a ds from v0 to vf and s0 to s0+Δs. This yields (1/2)(vf^2 − v0^2) = a Δs, which rearranges to vf^2 = v0^2 + 2 a Δs. The other forms use time (t) or mix in Δs in a way that doesn’t give a correct velocity-squared relation, and one would be dimensionally inconsistent if you tried to treat a Δs as a velocity.