How to find final velocity from acceleration and distance

Finding final velocity when you know acceleration and distance requires the velocity-displacement equation v² = u² + 2as, where you add twice the product of acceleration and distance to the square of initial velocity, then take the square root. For an object starting from rest (u = 0) that accelerates at 4 m/s² over 25 meters, calculate v² = 0 + 2(4)(25) = 200, so v = √200 = 14.14 m/s. Remember to take the square root as your final step—a common error is reporting v² as the velocity.

When initial velocity exists, include it in the calculation. A train traveling at 20 m/s that accelerates at 1.5 m/s² over 300 meters reaches v² = 400 + 2(1.5)(300) = 400 + 900 = 1300, giving v = 36.06 m/s. For deceleration with negative acceleration, the formula still applies: a skateboarder at 12 m/s who experiences -2 m/s² friction over 30 meters has v² = 144 + 2(-2)(30) = 144 - 120 = 24, so v = 4.90 m/s. The positive result confirms the skateboard hasn't stopped yet.

This formula proves invaluable when timing isn't available but distance markers and speed measurements are. Race engineers use it to determine corner exit speeds from acceleration data and track length. Safety analysts calculate impact velocities in crash tests by measuring deceleration distances. Always verify that the value under the square root is positive—if v² calculates to a negative number, it indicates the object stopped before reaching the specified distance, requiring a recalculation of actual stopping distance using 0 = u² + 2as rearranged to s = -u²/2a.