Motion with constant acceleration review

Review the key concepts, equations, and skills for motion with constant acceleration, including how to choose the best kinematic formula for a problem.

Key terms

TermMeaning
Kinematic variableVariable that describes the motion of an object over time. Includes displacement $\Delta x$ , time interval $t$, initial velocity $v_0$, final velocity $v$, and acceleration $a$.
Kinematic formulaFormula that describes the relationships between kinematic variables when acceleration is constant.

Equations

1. $v=v_0 + at$
2. $x = x_0 + v_0t + \dfrac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(x-x_0)$
4. $x - x_0 = \dfrac{1}{2}(v_0 + v)t$
Symbols
• $x_0$ is
• $x$ is the
• $t$ is the
• $v_0$ is initial velocity
• $v$ is final velocity
• $a$ is acceleration
Assumptions
• Acceleration is constant over the time interval

Using the kinematic formulas

Choosing the best kinematic formula

To choose the kinematic formula that's right for your problem, figure out which variable you are not given and not asked to find.
For example, we could use $v = v_0 + at$ to solve for the variables $v$, $v_0$, $a$, or $t$ if we knew the values of the other three variables. Note that each kinematic formula is missing one of the five kinematic variables.

Finding the known variables

Sometimes a known variable will not be explicitly given in a problem, but rather implied with codewords. For instance, "starts from rest" means $v_0=0$, "dropped" often means $v_0=0$, and "comes to a stop" means $v=0$.
Also, the magnitude of the acceleration due to gravity on all objects in free fall on Earth is usually assumed to be $g=9.8\dfrac{\text{m}}{\text{s}^2}$, so this acceleration will usually not be given explicitly.

Common mistakes and misconceptions

1. People forget that some of the kinematic variables are vectors and can have negative signs. For example, if upward is assumed to be positive, then the acceleration due to gravity must be negative: $a_g=-9.81\dfrac{\text{m}}{\text{s}^2}$. A missing negative sign is a very common mistake, so don't forget to check which direction is defined as positive!
2. People forget that the kinematic variables we plug into a kinematic formula must be consistent with that time interval. In other words, the initial velocity $v_0$ has to be the velocity of the object at the initial position and start of the time interval $t$. Similarly, the final velocity $v$ must be the velocity at the final position and end of the time interval $t$.
3. The second kinematic equation, $x=x_0+ v_0 t+\dfrac{1}{2}at^2$, might require using the
.