2.2 Velocity and Acceleration

Kinematics in One Dimension

Velocity and acceleration describe how objects move and how that motion changes over time. Together with the kinematic equations, they give you the tools to predict where something will be, how fast it'll be going, and when it'll get there.

Average vs. Instantaneous Velocity

Velocity is the rate of change of position with respect to time. Unlike speed, velocity is a vector quantity, meaning it has both magnitude and direction. A car going 60 m/s east has a different velocity than one going 60 m/s west.

Average velocity looks at the big picture of a trip:

$v_{avg} = \frac{\Delta x}{\Delta t}$

This uses total displacement (not total distance) divided by the total time interval. If you drive 150 km north in 2 hours, your average velocity is 75 km/h north, regardless of whether you sped up, slowed down, or stopped for gas along the way.

Instantaneous velocity is what your speedometer reads at a single moment:

$v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

Mathematically, it's the derivative of position with respect to time. You can think of it as the average velocity calculated over an infinitely small time interval.

Both are measured in m/s (SI units).

Concept of Acceleration

Acceleration is the rate of change of velocity with respect to time. It's also a vector quantity.

Average acceleration describes the overall change in velocity during a time interval:

$a_{avg} = \frac{\Delta v}{\Delta t}$

Instantaneous acceleration gives the acceleration at one specific moment:

$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Since velocity is already the derivative of position, acceleration is the second derivative of position:

$a = \frac{d^2x}{dt^2}$

Acceleration is measured in $\text{m/s}^2$.

A common point of confusion: "negative acceleration" does not always mean "slowing down." It means acceleration points in the negative direction. If an object is moving in the negative direction and has negative acceleration, it's actually speeding up. What matters is whether velocity and acceleration point the same way (speeding up) or opposite ways (slowing down).

Velocity and Acceleration Calculations

The four kinematic equations apply to motion with constant acceleration in one dimension. Each equation connects a different combination of five variables: $x$, $x_0$, $v$, $v_0$, $a$, and $t$.

1. $v = v_0 + at$

2. $x = x_0 + v_0 t + \frac{1}{2}at^2$

3. $v^2 = v_0^2 + 2a(x - x_0)$

4. $x = x_0 + \frac{1}{2}(v + v_0)t$

Notice that each equation is missing one of the five variables. That's the key to choosing the right one: pick the equation that leaves out the variable you neither know nor need.

Problem-solving steps:

1. Draw a quick diagram and define a positive direction.
2. List your known quantities ($x_0$, $v_0$, $a$, $t$, etc.) and identify the unknown you're solving for.
3. Select the kinematic equation that contains your unknown and your knowns.
4. Solve algebraically for the unknown, then plug in numbers.
5. Check that your units work out and that the answer is physically reasonable.

Two important special cases:

• Constant velocity ($a = 0$): The equations simplify to $x = x_0 + v_0 t$, since all acceleration terms drop out.
• Free fall ($a = g \approx 9.8 \text{ m/s}^2$ downward): The object accelerates due to gravity alone. If you define upward as positive, then $a = -9.8 \text{ m/s}^2$.

Graphs of Motion in One Dimension

Graphs are one of the most useful ways to analyze motion. The connections between position, velocity, and acceleration graphs all come down to slopes and areas.

Velocity-time (v-t) graphs:

• The slope at any point gives the acceleration at that moment.
• The area under the curve between two times gives the displacement during that interval.
• A horizontal line means constant velocity (zero acceleration).
• A straight line with positive slope means constant positive acceleration (velocity increasing steadily).
• A straight line with negative slope means constant negative acceleration (velocity decreasing steadily).

Acceleration-time (a-t) graphs:

• The area under the curve between two times gives the change in velocity ($\Delta v$) during that interval.
• A horizontal line means constant acceleration (like free fall, ignoring air resistance).

How these graphs connect:

The v-t graph is the integral of the a-t graph, and the a-t graph is the derivative of the v-t graph. The same derivative/integral relationship links the position-time graph to the v-t graph. So across all three graph types:

• Going from position to velocity to acceleration: take the slope (derivative).
• Going from acceleration to velocity to position: take the area under the curve (integral).

When reading any motion graph, look for intervals where the curve is flat (constant value), linearly increasing or decreasing (constant rate of change), or curved (changing rate of change). Identifying these regions tells you exactly what the object is doing during each phase of its motion.