When velocity changes, its rate of change is called acceleration. Mastering the three equations of uniformly accelerated motion is one of the central goals of introductory physics.
Basic Knowledge
Acceleration is the rate of change of velocity:a = Δv ÷ Δt = (v - v₀) ÷ t
Its unit is m/s².
Motion with constant acceleration is called uniformly accelerated linear motion, and the following three equations apply:
① v = v₀ + at (velocity)
② x = v₀t + ½at² (displacement)
③ v² = v₀² + 2ax (an equation without time)
Here, v₀ is the initial velocity, v is the velocity at time t, x is displacement, and a is acceleration.
If acceleration is negative (opposite to the initial velocity), the object slows down. If acceleration = 0, the motion is uniform linear motion (v = v₀ = constant).
Acceleration (the rate of change of velocity; unit: m/s²)
Uniformly accelerated linear motion (linear motion with constant acceleration)
Initial velocity v₀ (velocity at t=0)
v = v₀ + at (equation for finding velocity)
x = v₀t + ½at² (equation for finding displacement)
v² = v₀² + 2ax (equation without time t)
Further Study (Background and Significance)
The three equations can be derived from just two equations (v = v₀ + at and x = v₀t + ½at²). Equation ③ is obtained by eliminating t from equations ① and ②. Rather than memorizing the three equations, understand them as “① the relationship between v and t, ② the relationship between x and t, and ③ the relationship between x and v.” Then you will not be unsure which equation to use.
In a v-t graph (velocity-time graph), the slope represents acceleration, and the area enclosed by the graph and the horizontal axis represents displacement. It is a straight line for uniformly accelerated motion.
As Newton’s equation of motion F = ma shows, if the force is constant, acceleration is constant and uniformly accelerated motion occurs. Therefore, these three equations are fundamental to all of mechanics.
- Acceleration a = (v - v₀) / t; its unit is m/s².
- Equation ①: v = v₀ + at (relationship between v and t).
- Equation ②: x = v₀t + ½at² (relationship between x and t).
- Equation ③: v² = v₀² + 2ax (eliminates t).
- Slope of a v-t graph = acceleration; area = displacement.
- Negative acceleration = slowing down (accelerating in the opposite direction).
- The three equations can be derived from F=ma.
Points to Note (Common Confusions)
① Pay attention to the sign of acceleration. Decide the positive direction first, then determine the sign of a. If the object is slowing down, a is opposite to the initial velocity and is negative. ② Do not forget ½ in equation ②. x = v₀t + at² is incorrect. ③ Use equation ③ when time t is unknown or is not required. If time is known, equations ① and ② are usually quicker. ④ x is displacement, not distance from the starting point (it can be negative when the object returns).
Practice
- An object starts with an initial velocity of 0 m/s and accelerates at 4 m/s². Find its velocity after 3 seconds.
- For the motion above, find the displacement traveled in 3 seconds.
- An object moving at an initial velocity of 20 m/s slows down with an acceleration of -4 m/s². Find its displacement until its velocity becomes 0.