Which equation gives the final velocity when initial velocity, acceleration, and time are known?

Which equation gives the final velocity when initial velocity, acceleration, and time are known?

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

Answer: (C)

Velocity changes by the amount acceleration times time. Since acceleration is the rate of change of velocity, the change in velocity over a time interval t is a t, so the final velocity is v_f = v_0 + a t. This directly uses the known initial velocity, the constant acceleration, and the elapsed time, and it behaves correctly with sign: if acceleration points in the same direction as the motion, velocity increases; if opposite, it decreases. The other forms don’t fit this situation. A form that relates velocity to displacement (v_f^2 = v_0^2 + 2 a Δx) doesn’t involve time, so it’s not the right tool when time is known. An expression like v_f = v_0 + t / a is dimensionally inconsistent for velocity, since dividing time by acceleration doesn’t give velocity. A version with a minus sign would imply a specific deceleration direction regardless of how you define the sign of a, whereas the correct relation uses the plus with the actual (signed) acceleration.

Velocity changes by the amount acceleration times time. Since acceleration is the rate of change of velocity, the change in velocity over a time interval t is a t, so the final velocity is v_f = v_0 + a t. This directly uses the known initial velocity, the constant acceleration, and the elapsed time, and it behaves correctly with sign: if acceleration points in the same direction as the motion, velocity increases; if opposite, it decreases.

The other forms don’t fit this situation. A form that relates velocity to displacement (v_f^2 = v_0^2 + 2 a Δx) doesn’t involve time, so it’s not the right tool when time is known. An expression like v_f = v_0 + t / a is dimensionally inconsistent for velocity, since dividing time by acceleration doesn’t give velocity. A version with a minus sign would imply a specific deceleration direction regardless of how you define the sign of a, whereas the correct relation uses the plus with the actual (signed) acceleration.