Displacement, Velocity, and Acceleration

« Back to AP Physics Guide / Unit 1: Kinematics / Topic 1.2: Displacement, Velocity, and Acceleration

Conceptual visualization of a tangent line on a position-time graph translating into the exact value of a velocity-time graph.

In AP Physics C, motion is no longer limited to constant acceleration. Calculus is required to understand a continuously changing world.

If you took an algebra-based physics class prior to this, you spent weeks memorizing the “Big 4” kinematic equations. Here is the catch: those equations only work if acceleration is a perfectly constant number (like gravity, $g = 9.8 \text{ m/s}^2$ ).

But what if an object is attached to a spring, where the force gets stronger the further you pull it? Or what if a rocket is burning fuel and getting lighter, causing its acceleration to constantly increase? We must use the mathematics of continuous change: Calculus.

1. Kinematics as Derivatives (Moving Forward)

Instantaneous velocity is the rate of change of the object’s position with respect to time. Instantaneous acceleration is the rate of change of the object’s velocity. If you are given a position function $x(t)$ , you can find the velocity and acceleration functions simply by taking the derivative with respect to time $t$ .

$v_x = \frac{dx}{dt}$

$a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

Where $x$ is position, $v_x$ is instantaneous velocity, and $a_x$ is instantaneous acceleration.

Graphical Connection: The derivative represents the slope of the tangent line on a graph. The exact slope of a Position vs. Time graph at $t=2$ seconds is exactly equal to the velocity at $t=2$ seconds.

2. Kinematics as Integrals (Moving Backward)

If you are given an acceleration function $a(t)$ and need to find velocity or position, you cannot use derivatives. You must work backwards using Integration (Antiderivatives). Geometrically, integrating a function gives you the area under the curve of its graph.

$\Delta v_x = \int_{t_1}^{t_2} a_x(t) dt$

$\Delta x = \int_{t_1}^{t_2} v_x(t) dt$

Notice the $\Delta$ symbol! A definite integral gives you the change in the value, not the absolute final value.

⚠️ The Initial Value Trap: When taking an indefinite integral, you must always add a constant $+ C$ . In kinematics, this constant represents your starting conditions: Initial Velocity ( $v_0$ ) or Initial Position ( $x_0$ ). Do not forget to add them to your final equation!

3. Quick AP Practice

📚 Topic 1.2 Mastery Challenge

1. An object’s position is given by $x(t) = 4t^2 - 12t$ . At what time $t$ does the object instantaneously reverse its direction of motion?

Check Answer

An object reverses direction when its velocity crosses zero. First, find the velocity function by taking the derivative of position:
$v(t) = \frac{dx}{dt} = 8t - 12$ .

Set velocity equal to zero:
$0 = 8t - 12 \Rightarrow 8t = 12 \Rightarrow t = \mathbf{1.5 \text{ s}}$ .

2. The acceleration of a particle is given by $a(t) = 6t$ . If the particle starts from rest at an initial position of $x = 5 \text{ m}$ , what is its position at $t = 2 \text{ s}$ ?