Constant-Acceleration Motion Equations Study Pack

What Constant Acceleration Means Physically

• •Constant acceleration means the velocity changes by the same amount every second — a car accelerating uniformly from rest gains the same increment of speed in each equal time interval.
• •If acceleration varies with time (e.g., a rocket burning fuel), these equations give incorrect results and calculus-based methods are required instead.
• •Zero acceleration is a valid special case: setting a = 0 in any kinematic equation reduces it to uniform motion, where displacement = v₀t.

The Five Kinematic Variables

• •Every constant-acceleration problem in one dimension involves five variables: initial velocity v₀, final velocity v, acceleration a, displacement Δx, and elapsed time t.
• •A complete problem provides three of these five variables; the kinematic equations allow you to solve for the remaining two.
• •Displacement Δx represents the net change in position (final minus initial), which is distinct from total distance traveled when an object reverses direction.

• •Equation 1: Velocity as a Function of Time — v = v₀ + at
• •Constant acceleration a is defined as the change in velocity divided by time: a = (v − v₀)/t.
• •Rearranging that definition directly yields v = v₀ + at, making this equation a restatement of what constant acceleration means.
• •This equation is most useful when displacement is not needed and time is either given or sought.
• •Equation 2: Displacement Using Average Velocity — Δx = ½(v₀ + v)t
• •Because velocity changes linearly under constant acceleration, the average velocity over any interval equals exactly (v₀ + v)/2.
• •Multiplying this average velocity by elapsed time gives displacement: Δx = ½(v₀ + v)t.
• •This equation is particularly efficient when acceleration is not explicitly given but initial and final velocities are both known.
• •Equation 3: Displacement as a Function of Time — Δx = v₀t + ½at²
• •Substituting Equation 1 (v = v₀ + at) into Equation 2 and simplifying produces Δx = v₀t + ½at².
• •The v₀t term represents the displacement the object would have covered at constant initial velocity; the ½at² term is the additional displacement contributed by acceleration.
• •Geometrically, v₀t is the rectangular area under the velocity-time graph and ½at² is the triangular area above it — together they give total area, which equals displacement.
• •Equation 4: Velocity as a Function of Displacement — v² = v₀² + 2aΔx
• •Eliminating t algebraically from Equations 1 and 2 produces v² = v₀² + 2aΔx.
• •This equation is the most efficient choice when elapsed time is neither given nor needed, such as finding the speed of an object after it falls a known height.
• •The equation also reveals that displacement and the change in v² are directly proportional when acceleration is fixed.