Displacement, Velocity, and Acceleration - AP Physics C: Mechanics Study Guide

Getting Started

To analyze the motion of any object, from a planet to a subatomic particle, we must first establish a language to describe it. We will model objects as point particles, focusing on their location in space and how that location changes over time. The core question of kinematics is: how can we use the tools of calculus to precisely define and relate an object's position, velocity, and acceleration at any given moment?

What You Should Be Able to Do

After working through this section, you should be able to:

• Calculate the displacement vector of a particle given its initial and final position vectors.

• Determine the average velocity over a time interval and distinguish it from instantaneous velocity.

• Find the instantaneous velocity vector, $\vec{v}(t)$, by taking the time derivative of the position vector, $\vec{r}(t)$.

• Find the instantaneous acceleration vector, $\vec{a}(t)$, by taking the time derivative of the velocity vector, $\vec{v}(t)$.

• Reconstruct velocity from acceleration, or position from velocity, using definite integration.

Key Concepts & Mechanisms

The language of motion is built upon the concept of change over time. We begin with a baseline state—position—and use the principles of differential calculus to describe how that state evolves.

• Baseline State: Position

The fundamental state of a particle is its position, a vector that specifies its location relative to a chosen origin. It is a function of time, denoted as $\vec{r}(t)$. In Cartesian coordinates, this can be written as $\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, where $x(t)$, $y(t)$, and $z(t)$ are scalar functions describing the particle's coordinates at time $t$. The change in this baseline state is called displacement, $\Delta \vec{r}$, defined as the difference between the final and initial position vectors: $\Delta \vec{r}={\vec{r}}_{final}-{\vec{r}}_{initial}$.

• Differential Driver(s): Rates of Change

The evolution of the position state is driven by two key rates of change:

1. Velocity as the Rate of Change of Position:

   The first driver of motion is velocity. The average velocity, ${\vec{v}}_{avg}$, describes the overall rate of displacement over a finite time interval, $\Delta t$:

   ${\vec{v}}_{avg}=\frac{\Delta \vec{r}}{\Delta t}=\frac{\vec{r}(t_f)-\vec{r}(t_i)}{t_f-t_i}$

   To find the velocity at a single moment, we take the limit as the time interval shrinks to zero. This defines the instantaneous velocity, $\vec{v}(t)$, as the time derivative of the position vector:

   $\vec{v}(t)={\lim }_{\Delta t\to 0}\frac{\Delta \vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}$

2. Acceleration as the Rate of Change of Velocity:

   The second driver of motion is acceleration, which describes how the velocity itself is changing. The average acceleration, ${\vec{a}}_{avg}$, is the rate of change of velocity over a time interval:

   ${\vec{a}}_{avg}=\frac{\Delta \vec{v}}{\Delta t}=\frac{\vec{v}(t_f)-\vec{v}(t_i)}{t_f-t_i}$

   The instantaneous acceleration, $\vec{a}(t)$, is the time derivative of the velocity vector, representing the rate of change of velocity at a specific instant:

   $\vec{a}(t)={\lim }_{\Delta t\to 0}\frac{\Delta \vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}$

   Since velocity is the derivative of position, acceleration is the second derivative of position with respect to time: $\vec{a}(t)=\frac{d}{dt}\left(\frac{d\vec{r}}{dt}\right)=\frac{d^2\vec{r}}{d{t}^2}$.

• Integral Relationships

The Fundamental Theorem of Calculus allows us to reverse this process. If we know the rate of change (e.g., acceleration), we can find the net change in the original function (e.g., velocity) by integration. The change in velocity from an initial time $t_0$ to a final time $t$ is the definite integral of the acceleration function. Likewise, the displacement is the definite integral of the velocity function.

$\Delta \vec{v}=\vec{v}(t)-\vec{v}(t_0)={\int }_{t_0}^t\vec{a}(t^{\prime })d{t}^{\prime }$

$\Delta \vec{r}=\vec{r}(t)-\vec{r}(t_0)={\int }_{t_0}^t\vec{v}(t^{\prime })d{t}^{\prime }$

These integral relationships are essential for predicting the future state of a system when its acceleration or velocity history is known.

Key Models & Diagrams

The relationships between position, velocity, and acceleration form a clear hierarchy based on calculus. This can be visualized as a "calculus ladder," where moving down the ladder involves differentiation and moving up involves integration.

Key Components & Evidence

• Position ($\vec{r}$): A vector quantity describing an object's location relative to an origin. Its SI unit is the meter (m).

• Time ($t$): The independent scalar variable against which motion is measured. Its SI unit is the second (s).

• Displacement ($\Delta \vec{r}$): The vector change in position, ${\vec{r}}_{final}-{\vec{r}}_{initial}$. It is independent of the path taken.

• Average Velocity (${\vec{v}}_{avg}$): The ratio of displacement to the time interval, $\Delta \vec{r}/\Delta t$. It represents the constant velocity that would produce the same displacement in the same time.

• Instantaneous Velocity ($\vec{v}$): The time derivative of position, $d\vec{r}/dt$. This vector is always tangent to the object's path. Its magnitude is the instantaneous speed.

• Instantaneous Acceleration ($\vec{a}$): The time derivative of velocity, $d\vec{v}/dt$. This vector describes the rate of change of the velocity vector in both magnitude and direction.

• Derivative ($d/dt$): The mathematical operator that yields the instantaneous rate of change of a function.

• Integral ($\int dt$): The mathematical operator that accumulates the total change in a quantity from its rate of change.

• Point Particle Model: An assumption that treats an object as a dimensionless point, allowing us to describe its motion without considering its size, shape, or rotation.

Skill Snapshots

• Derivational Relationships:

  • Driver: A position function, e.g., $x(t)=A\sin (\omega t)$. → Change: The velocity is found by differentiation: $v(t)=\frac{dx}{dt}=A\omega \cos (\omega t)$.

  • Driver: A velocity vector, e.g., $\vec{v}(t)=(3t^2)\hat{i}+(4)\hat{j}$. → Change: The acceleration is found by differentiation: $\vec{a}(t)=\frac{d\vec{v}}{dt}=(6t)\hat{i}$.

  • Driver: A non-zero acceleration. → Change: The velocity vector is changing over time.

• Comparison:

  • Displacement vs. Distance: For a trip from home to school and back, your final displacement is zero, but the distance traveled is twice the one-way distance.

  • Average vs. Instantaneous Velocity: On a road trip, your average velocity might be 60 mph, but your instantaneous velocity on the speedometer varies continuously and is zero when you stop.

  • Velocity vs. Speed: A car traveling at a constant speed of 20 m/s around a circular track has a constant speed, but its velocity vector is continuously changing direction, implying a non-zero acceleration.

• Change and Continuity Over Time:

  • Baseline: An object's location is specified by its position function, $\vec{r}(t)$.

  • Change 1: If the velocity function $\vec{v}(t)$ is non-zero for an interval, the object's position changes during that interval.

  • Change 2: If the acceleration function $\vec{a}(t)$ is non-zero, the object's velocity changes, meaning it is speeding up, slowing down, or changing direction.

  • Continuity: If acceleration is zero, velocity is constant. This state of constant velocity (including $\vec{v}=0$) will persist indefinitely until an acceleration is introduced.

Common Misconceptions & Clarifications

1. Misconception: An object with zero velocity must have zero acceleration.

   • Clarification: Acceleration is the rate of change of velocity. At the highest point of its trajectory, a vertically thrown ball has an instantaneous velocity of zero, but its velocity is actively changing (from upward to downward), so its acceleration is non-zero (specifically, the acceleration due to gravity).

2. Misconception: Acceleration and velocity must point in the same direction.

   • Clarification: Acceleration points in the direction of the change in velocity ($\Delta \vec{v}$). If an object is slowing down, its acceleration vector is in the opposite direction of its velocity vector. In uniform circular motion, acceleration is perpendicular to velocity.

3. Misconception: Displacement is the same as distance traveled.

   • Clarification: Displacement is a vector from the starting point to the ending point ($\Delta \vec{r}={\vec{r}}_f-{\vec{r}}_i$). Distance is a scalar representing the total length of the path traveled. You can run a full 400 m lap on a track, covering a distance of 400 m, but your displacement will be zero.

4. Misconception: Average velocity is calculated as $({\vec{v}}_i+{\vec{v}}_f)/2$.

   • Clarification: This formula is a special case that is only valid for motion with constant acceleration. The universal definition, which is always correct, is ${\vec{v}}_{avg}=\Delta \vec{r}/\Delta t$.

One-Paragraph Summary

Kinematics provides the mathematical framework for describing motion by defining position, velocity, and acceleration as vector functions of time. The core of this framework lies in calculus: instantaneous velocity is the time derivative of position, and instantaneous acceleration is the time derivative of velocity. This hierarchical relationship implies that acceleration is the second time derivative of position. Conversely, integration allows for the reconstruction of motion; velocity can be found by integrating acceleration over time, and displacement by integrating velocity. By distinguishing between average quantities over intervals and instantaneous quantities at a point in time, this model provides a precise and powerful language to analyze the motion of any object treated as a point particle.