How can I find displacement from initial velocity, constant acceleration, and elapsed time?

For one-dimensional motion with constant acceleration, use:

Δx = v₀t + ½at²

Here, Δx is displacement, v₀ is initial velocity, a is acceleration, and t is elapsed time. Substitute the values with their signs and use compatible units. The equation assumes acceleration stays constant throughout the interval. [S1]

Choose a positive direction first. Then represent velocities and acceleration along that axis, including negative signs when they point in the opposite direction. The result is signed displacement along the chosen axis.

Acceleration multiplied by time gives a change in velocity, not a displacement. During the interval, the velocity changes, and that change contributes to displacement as time passes. For constant acceleration, the resulting contribution is ½at². The factor of ½ comes from using average velocity over the interval: with constant acceleration, velocity changes steadily from its initial value. [S1]

A useful way to separate the terms is that v₀t is the displacement contribution from the initial velocity, while ½at² accounts for the additional contribution from the acceleration. Both terms represent length, even though acceleration itself is measured in length per time squared. [S1]

Suppose an object has initial velocity v₀ = 3 m/s, constant acceleration a = 2 m/s², and elapsed time t = 4 s. Let the positive direction be the direction of its initial motion.

1. Substitute into the equation: Δx = (3 m/s)(4 s) + ½(2 m/s²)(4 s)².
2. Calculate the first term: (3 m/s)(4 s) = 12 m.
3. Calculate the acceleration term: ½ × 2 × 16 = 16, with units (m/s²)(s²) = m.
4. Add the terms: Δx = 12 m + 16 m = 28 m.

Dimensional check: (m/s)(s) reduces to m, and (m/s²)(s²) also reduces to m. Both terms have units of length, so they can be added. [S1]

The example uses positive acceleration in the chosen positive direction. If your input values point in different directions, keep their signs in the substitution rather than treating every value as positive.

If an object is moving in the positive direction and slowing down, its acceleration points opposite that direction. With that axis choice, use a negative value for a; the same equation applies with signed quantities. [S1]

Make the time units consistent before calculating. For example, when velocity is in m/s and acceleration is in m/s², express elapsed time in seconds. Then use that time in both terms, squaring it in ½at². Keep units attached as you work, and check that each term reduces to a length.

Reusable setup:

• Positive direction: [state direction]
• Initial velocity: v₀ = [value and unit]
• Acceleration: a = [signed value and unit]
• Elapsed time: t = [value in matching time unit]
• Calculation: Δx = v₀t + ½at² = [first term] + [second term] = [result and unit]
• Check: both terms have units of length.