Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion. It focuses on describing how objects move in terms of displacement, velocity, acceleration, and time. Understanding kinematics is essential because it provides the foundation for later topics in AP Physics 1, such as dynamics and energy. In this unit, we learn to analyze motion using words, diagrams, equations, and graphs.
Distance and displacement are two fundamental ways to describe how far and in what manner an object moves. While both measure motion, distance refers only to the total path length traveled, whereas displacement focuses on the change in position from start to finish. This distinction is important in physics because displacement includes direction, which affects how we calculate velocity and acceleration. In real-world applications, distance might tell you how far you ran, but displacement tells you your overall change in position, which can sometimes be much less.
All physical quantities fall into two categories: scalars and vectors. Scalars are quantities that have only magnitude, such as mass or temperature, while vectors have both magnitude and direction, such as velocity or displacement. This distinction is important because the rules for combining them differ. Scalars combine through normal arithmetic, while vectors require vector addition, often represented graphically using arrows or mathematically using trigonometry. Understanding the difference helps students avoid common mistakes, such as treating speed and velocity as the same thing.
Distance and displacement are two fundamental ways to describe how far and in what manner an object moves. Distance measures the total path length traveled, while displacement focuses on the straight-line change in position from start to finish. This distinction is crucial because physics often requires knowing not just how far something moved, but also in what direction. For example, a runner completing a full lap on a track covers a large distance, but their displacement is zero because they end up back where they started. This is why AP Physics 1 emphasizes displacement more heavily than distance.
To describe motion in physics, we use three connected quantities: position, velocity, and acceleration. Position tells us where an object is relative to a chosen reference point. Velocity describes how quickly and in what direction the position is changing. Acceleration shows how the velocity itself is changing over time. These three ideas are the backbone of kinematics, and they are often represented graphically using position-time, velocity-time, and acceleration-time graphs. Understanding how these graphs relate to each other is essential for solving AP Physics 1 problems.
Position is the specific location of an object relative to a reference point, such as the origin of a coordinate system. It is a vector quantity because it includes both distance and direction. Position is measured in units of meters (m) and represented as \(x\) in one-dimensional motion. A positive position means the object is located in the positive direction from the origin, while a negative position means it is located in the opposite direction. In AP Physics 1, position is often analyzed using position-time graphs, where the slope of the graph provides useful information about the object’s velocity.
Velocity is the rate of change of displacement with respect to time. Unlike speed, which only tells you how fast you are moving, velocity tells you both how fast and in what direction. It is a vector quantity, meaning it can be positive or negative depending on direction. The SI unit of velocity is meters per second (\(m/s\)). In a position-time graph, the slope represents the object’s velocity. An object moving with constant velocity will have a straight-line graph, while an object changing velocity will have a curved graph. Average velocity differs from instantaneous velocity, which is the velocity at a specific instant in time.
Acceleration is the rate at which velocity changes with respect to time. Because it depends on changes in velocity, acceleration is also a vector quantity. It can occur when an object speeds up, slows down, or changes direction. The SI unit of acceleration is meters per second squared (\(m/s^2\)). In AP Physics 1, acceleration is especially important in analyzing free-fall motion, where objects accelerate downward at approximately \(9.8\ m/s^2\) near Earth’s surface. On a velocity-time graph, the slope of the line represents acceleration. Constant acceleration produces a straight-line graph, while changing acceleration produces a curved graph.
Graphs are powerful tools in kinematics because they show how position, velocity, and acceleration are related. A position-time graph’s slope reveals velocity, while a velocity-time graph’s slope reveals acceleration. Conversely, the area under a velocity-time graph gives displacement, and the area under an acceleration-time graph gives the change in velocity. This makes graphs essential for problem solving because they provide both geometric and analytical ways to interpret motion. Mastering how to read and interpret these graphs is a critical AP Physics 1 skill.
Although the words “speed” and “velocity” are often used interchangeably in everyday language, in physics they have very different meanings. Speed is a scalar quantity, meaning it only tells us how fast something is moving without any regard for direction. Velocity, on the other hand, is a vector quantity that tells us both how fast an object is moving and in what direction. This distinction is extremely important in kinematics because two objects can have the same speed but very different velocities if they are traveling in opposite directions. Understanding the difference helps students avoid major mistakes when analyzing motion.
Speed describes the rate at which an object covers distance. Since it does not include direction, it cannot tell us whether the object is moving forward, backward, or in a circle. The SI unit of speed is meters per second (\(m/s\)), and it is usually calculated as the total distance traveled divided by the time it takes to travel that distance. While useful for everyday situations like driving, speed alone is insufficient for many physics problems because it ignores directional information that velocity includes.
Velocity describes the rate of change of displacement over time, making it a vector quantity. Unlike speed, velocity includes both magnitude and direction, so it can be positive or negative depending on which way the object is moving. Instantaneous velocity is the velocity at a single moment, while average velocity considers the total displacement divided by total time. The SI unit of velocity is meters per second (\(m/s\)). On a position-time graph, the slope represents velocity, while on a velocity-time graph, a horizontal line indicates constant velocity. Velocity is crucial in AP Physics 1 because it determines not just how fast something moves, but also how its position changes relative to a reference point.
Although speed and velocity are related, they are not interchangeable. Speed only measures how fast an object is moving, while velocity includes both speed and direction. This means two objects can have the same speed but very different velocities if they are traveling in different directions. On a position-time graph, the slope gives velocity (with direction), while speed is always considered positive. In AP Physics problems, it is essential to pay attention to whether the question is asking for speed or velocity because the answer may change depending on direction.
Acceleration is one of the most important concepts in kinematics because it tells us how velocity changes over time. While velocity measures how fast and in what direction an object is moving, acceleration measures how quickly that velocity increases, decreases, or changes direction. Acceleration is a vector quantity, meaning it has both magnitude and direction. It can be positive (speeding up in the positive direction), negative (slowing down in the positive direction, often called deceleration), or even involve changing direction without changing speed, such as when a car turns around a curve. The SI unit for acceleration is meters per second squared (\( m/s^2 \)), which means that for every second that passes, the velocity changes by a certain number of meters per second. Understanding acceleration allows us to predict motion in situations like free fall, car braking, and projectile motion.
It is important to understand that positive acceleration does not always mean “speeding up,” and negative acceleration does not always mean “slowing down.” Whether the object speeds up or slows down depends on the relationship between the velocity’s direction and the acceleration’s direction. If acceleration and velocity point in the same direction, the object speeds up. If they point in opposite directions, the object slows down. This concept is especially important when interpreting velocity-time graphs in AP Physics 1.
Uniform acceleration means that an object’s acceleration remains constant throughout the motion. This type of motion is especially important in AP Physics 1 because it allows us to use the kinematic equations to predict displacement, velocity, and time. A common real-world example of uniform acceleration is free fall near Earth’s surface, where the acceleration due to gravity is approximately \( 9.8\ m/s^2 \) downward. When acceleration is uniform, velocity changes by equal amounts in equal time intervals, and position-time graphs form parabolas while velocity-time graphs form straight lines.
Graphs provide a powerful way to visualize acceleration. On a velocity-time graph, the slope of the line gives the acceleration. If the line slopes upward, acceleration is positive; if it slopes downward, acceleration is negative. The area under the velocity-time graph gives displacement. On an acceleration-time graph, the area under the curve gives the change in velocity. These relationships allow us to extract motion details even without direct equations, which is a key skill on the AP Physics 1 exam.
Free fall is one of the most important examples of uniformly accelerated motion in AP Physics 1. It refers to the motion of any object that is moving only under the influence of gravity, with no other forces like air resistance acting significantly. In free fall near Earth’s surface, all objects experience the same constant acceleration downward, regardless of their mass. This acceleration, denoted as \( g \), has a magnitude of approximately \( 9.8\ m/s^2 \). This means that every second, the object’s velocity increases (in magnitude) by about 9.8 meters per second. Understanding free fall helps students see how acceleration, velocity, and displacement interact in real-world situations, such as dropping an object, throwing it upward, or analyzing projectile motion.
Students often believe that heavier objects fall faster than lighter ones. In reality, ignoring air resistance, all objects accelerate at the same rate under gravity. Another misconception is that acceleration is zero at the top of a projectile’s path since the velocity is zero at that instant. However, acceleration remains constant and directed downward throughout the motion. Recognizing and correcting these misconceptions is essential for success on AP Physics 1 problems.
Scenario: A ball is thrown straight up with an initial velocity of \( 20\ m/s \). How high does it go, and how long until it returns to the thrower’s hand (ignoring air resistance)?
Answer: The ball reaches a maximum height of about 20.4 m and returns after about 4.08 s.
The kinematic equations are a set of formulas used to describe the motion of objects under constant acceleration. They connect displacement, velocity, acceleration, and time, making them powerful tools for solving motion problems in AP Physics 1. These equations only apply when acceleration is uniform (constant). If acceleration is changing, other methods such as calculus or graphical analysis are required. Mastering these equations is essential because they appear in nearly every kinematics unit problem, especially those involving free fall, uniformly accelerated motion, and one-dimensional motion.
Displacement with Average Velocity:
\[
\Delta x = \frac{v + v_0}{2} \cdot t
\]
This equation calculates displacement when the initial and final velocities are known. It is especially useful when acceleration is constant but not directly given.
Displacement with Initial Velocity and Acceleration:
\[
\Delta x = v_0 t + \frac{1}{2} a t^2
\]
This formula is often used when the object starts with a known initial velocity and moves under constant acceleration for a given time.
Final Velocity from Initial Velocity and Acceleration:
\[
v = v_0 + a t
\]
This equation provides the final velocity after accelerating for a certain time. It is one of the simplest and most commonly used in motion analysis.
Final Velocity and Displacement:
\[
v^2 = v_0^2 + 2 a \Delta x
\]
This version eliminates time, making it especially useful when displacement and acceleration are known, but time is not provided.
These equations only apply when acceleration is constant.
Symbols:
To master kinematics, it is essential not only to know the equations but also to practice applying them. Below are worked examples demonstrating how each of the four main kinematic equations is used. Notice how choosing the correct equation depends on which variables are known and which are unknown. Always begin by identifying the known quantities, choosing a coordinate system, and writing down what you are solving for.
Problem: A car accelerates from 10 m/s to 30 m/s in 5 seconds. How far does it travel in this time?
Problem: A ball is thrown upward with an initial velocity of 15 m/s. How high does it rise after 1.5 seconds?
Problem: A motorcycle starts at rest and accelerates at 4 m/s² for 8 seconds. What is its final velocity?
Problem: A car moving at 25 m/s comes to a stop after skidding 50 meters. What was the car’s acceleration?
Projectile motion is a type of two-dimensional motion where an object is launched into the air and moves under the influence of gravity alone (ignoring air resistance). The motion can be broken down into two independent components: horizontal and vertical. Horizontally, the object moves with constant velocity since there is no horizontal acceleration. Vertically, the object accelerates downward at \( g = 9.8\ m/s^2 \). By analyzing these components separately, we can predict the object’s entire trajectory. Angled motion occurs when a projectile is launched at an angle instead of straight up or straight across, requiring both horizontal and vertical motion to be considered together.
The horizontal motion of a projectile is simple because there is no horizontal acceleration (ignoring air resistance). This means the horizontal velocity remains constant throughout the flight. The horizontal displacement, often called range, depends only on the initial horizontal velocity and the time the projectile is in the air.
Equation: \[ x = v_{x} \cdot t \] where \( v_{x} \) = horizontal velocity and \( t \) = total time of flight.
Vertical motion in projectile problems is identical to free fall. The projectile accelerates downward at a constant rate of \( 9.8\ m/s^2 \), regardless of whether it is rising or falling. This means vertical velocity decreases on the way up, reaches zero at the top, and then increases downward on the way down.
Equations (using kinematics):
\( y = v_{y0} t + \frac{1}{2} a t^2 \)
\( v_y = v_{y0} + at \)
\( v_y^2 = v_{y0}^2 + 2a y \)
Three important quantities in projectile motion are the total time in the air (time of flight), the maximum height reached, and the horizontal range. Each of these can be derived using the kinematic equations with vertical and horizontal components separated. Remember that the total flight time depends only on vertical motion, while horizontal range depends on both time and horizontal velocity.
Time of Flight: \[ t = \frac{2 v_0 \sin \theta}{g} \] (valid when the projectile lands at the same height it was launched).
Maximum Height: \[ H = \frac{(v_0 \sin \theta)^2}{2g} \]
Range: \[ R = \frac{v_0^2 \sin(2\theta)}{g} \] (valid when landing height = launch height).
Angled projectile motion occurs when an object is launched with an initial velocity at some angle \( \theta \) above the horizontal. Unlike horizontal launches, where the object starts with only a horizontal velocity, angled launches have both horizontal and vertical components right from the start. The horizontal component determines how far the projectile travels, while the vertical component determines how long the projectile remains in the air and how high it rises. Together, these components create a parabolic path. Mastering angled projectile motion is essential for AP Physics 1 because it combines one-dimensional kinematics with vector decomposition.
Projectile motion is symmetrical when launch and landing occur at the same height. This means the time going up equals the time coming down, and the vertical speed at launch equals the vertical speed just before landing (but in the opposite direction). The horizontal distance covered in each half of the flight is the same, so the midpoint of the flight corresponds to maximum height. Understanding this symmetry allows students to quickly check answers and make predictions about motion without always doing full calculations.
The launch angle strongly influences how far and how high a projectile travels. A low angle produces a shorter, flatter path, while a high angle produces a taller, shorter-range path. The maximum horizontal range occurs when the projectile is launched at 45°, assuming level ground and no air resistance. Complementary angles (like 30° and 60°) produce the same range but different flight times and maximum heights.
In physics, motion is always described relative to a chosen reference frame. A reference frame is simply a perspective from which motion is measured, usually defined by a coordinate system and a clock. The same object can have different velocities when observed from different frames of reference. For example, a passenger walking inside a moving train appears to move slowly when observed from inside the train, but an outside observer sees the passenger moving much faster because they combine the train’s velocity with the passenger’s velocity. Understanding reference frames is critical in AP Physics 1 because many problems require analyzing motion from different perspectives.
Relative velocity is calculated using a simple vector equation. If object A moves relative to object B, and B moves relative to the ground, then the velocity of A relative to the ground is:
\[ v_{A/G} = v_{A/B} + v_{B/G} \]
Many real-world motions cannot be described with a single straight line, so AP Physics 1 expands kinematics to two dimensions. In two-dimensional motion, vectors are used to describe quantities like displacement, velocity, and acceleration. Each vector can be broken down into horizontal (x) and vertical (y) components, which are treated independently. The key principle is that horizontal and vertical motions do not affect each other, except for sharing the same time of motion. Understanding how to add, subtract, and resolve vectors is essential for problems involving projectile motion, angled throws, and relative motion.