Kinematics is the language physicists use to describe how objects move without yet worrying about the forces that cause the motion. In this unit you learn to track an object's position over time and to connect that position to its velocity and acceleration. Mastering these tools, including motion graphs and the constant-acceleration equations, sets the foundation for everything that follows in AP Physics 1.
Scalars versus vectors
Every quantity in kinematics is either a scalar or a vector. A scalar has only a magnitude, a single number with units, such as 5 m of distance or 12 m/s of speed. A vector has both magnitude and direction, such as a displacement of 5 m east or a velocity of 12 m/s north. In one-dimensional motion you can represent vector direction with a simple sign: positive for one direction along your chosen axis and negative for the other. Choosing a positive direction at the start of a problem and sticking with it is one of the most important habits in this unit, because mixing up signs is the most common source of kinematics errors. When you add vectors, direction matters, so two 3 m displacements can combine into anything from 0 m to 6 m depending on their directions.
Position, displacement, and distance
Position, written x, tells you where an object is relative to a chosen origin along an axis. Displacement is the change in position, delta x = x_final - x_initial, and it is a vector pointing from the start to the end of the trip. Distance is the total length of path actually traveled and is a scalar that can never be negative. These two ideas often differ. If you walk 4 m east and then 4 m back west, your distance traveled is 8 m but your displacement is 0 m because you ended where you started. Only when motion stays in a single direction without reversing does the magnitude of displacement equal the distance.
Velocity and speed
Average velocity is displacement divided by the time interval, v_avg = delta x / delta t, and it is a vector that carries the sign of the displacement. Average speed is total distance divided by total time and is always a positive scalar. Instantaneous velocity is the velocity at a single instant, which graphically is the slope of the position versus time graph at that point. Instantaneous speed is just the magnitude of instantaneous velocity, which is what a car's speedometer reads. Because of the displacement versus distance distinction, average velocity and average speed can have very different values for the same trip, but instantaneous speed always equals the size of instantaneous velocity.
Acceleration
Acceleration is the rate at which velocity changes, a_avg = delta v / delta t, with units of m/s^2. It too is a vector. A crucial point is that acceleration describes change in velocity, not whether an object is moving fast. An object can be moving quickly yet have zero acceleration if its velocity is constant, and an object momentarily at rest, like a ball at the top of its flight, can still be accelerating. When acceleration points in the same direction as velocity the object speeds up; when it points opposite the velocity the object slows down. The sign of acceleration alone does not tell you whether something is speeding up or slowing down, only the relationship between the signs of velocity and acceleration does.
Interpreting motion graphs
Three graphs describe the same motion and reading them fluently is a tested skill. On a position versus time graph, the slope at any point is the instantaneous velocity, so a steeper line means faster motion and a horizontal line means the object is at rest. On a velocity versus time graph, the slope is the acceleration, and the area between the line and the time axis equals the displacement. On an acceleration versus time graph, the area under the line gives the change in velocity. A curved position graph signals acceleration, while a straight, sloped velocity graph signals constant acceleration. Practice converting between the three: a constant positive velocity becomes a straight diagonal on a position graph, a horizontal line on a velocity graph, and a line sitting on zero on an acceleration graph.
Kinematic equations for constant acceleration
When acceleration is constant, four equations relate the five quantities of displacement (delta x), initial velocity (v_0), final velocity (v), acceleration (a), and time (t). They are: v = v_0 + a*t; delta x = v_0*t + (1/2)*a*t^2; v^2 = v_0^2 + 2*a*delta x; and delta x = (1/2)*(v_0 + v)*t. Each equation leaves out exactly one variable, so your strategy is to list what you know, identify the unknown you want, and pick the equation missing the variable you neither know nor need. Worked example: a car starts from rest (v_0 = 0) and accelerates at 3 m/s^2 for 5 s. How far does it travel? Use delta x = v_0*t + (1/2)*a*t^2 = 0*(5) + (1/2)*(3)*(5)^2 = (1/2)*(3)*(25) = 37.5 m. Its final velocity is v = v_0 + a*t = 0 + 3*5 = 15 m/s.
Free fall
Free fall is motion under gravity alone, with air resistance ignored. Near Earth's surface every freely falling object has the same downward acceleration of magnitude g, approximately 9.8 m/s^2 (often rounded to 10 m/s^2 for quick estimates). This acceleration is the same for a feather and a hammer in a vacuum, and it acts whether the object is moving up, moving down, or momentarily at rest. A ball thrown straight up slows as it rises, stops for an instant at the top where velocity is zero but acceleration is still g downward, then speeds up as it falls. The kinematic equations apply directly with a set equal to g; just be consistent about whether you call downward positive or negative.
Projectile motion
A projectile is launched and then moves under gravity alone in two dimensions. The key insight is that the horizontal and vertical motions are independent and can be analyzed separately while sharing the same clock. Horizontally, with no force acting, the velocity is constant, so x = v_x*t. Vertically, the motion is free fall with acceleration g, so the y equations are the constant-acceleration equations with a = g. The launch velocity is split into components: v_x = v*cos(theta) and v_y = v*sin(theta). Because the two directions are independent, a bullet fired horizontally and one simply dropped from the same height hit the ground at the same time. Time of flight is found from the vertical motion and then used in the horizontal equation to get range.
Relative velocity
Velocity is always measured relative to some reference frame, and different observers can record different velocities for the same object. To find the velocity of object A relative to object B, you add velocity vectors: v_AB = v_A - v_B, where each velocity is measured relative to the ground. In one dimension this is straightforward sign arithmetic. For example, if a person walks at 2 m/s forward on a train moving at 15 m/s, the person's velocity relative to the ground is 17 m/s, but relative to the train it is just 2 m/s. In two dimensions, such as a boat crossing a river, you add the boat's velocity relative to the water and the water's velocity relative to the ground as vectors to find the boat's velocity relative to the ground.
Key terms
Scalar
A quantity described by magnitude only, with no direction, such as distance, speed, or time.
Vector
A quantity with both magnitude and direction, such as displacement, velocity, or acceleration.
Position
The location of an object relative to a chosen origin along an axis, written x.
Displacement
The vector change in position, delta x = x_final - x_initial, independent of the path taken.
Distance
The total length of path traveled, a non-negative scalar.
Average velocity
Displacement divided by time interval, a vector: v_avg = delta x / delta t.
Instantaneous velocity
The velocity at a single instant, equal to the slope of a position versus time graph.
Speed
The magnitude of velocity, a non-negative scalar.
Acceleration
The rate of change of velocity, a vector with units m/s^2: a = delta v / delta t.
Free fall
Motion under gravity alone, with acceleration g of about 9.8 m/s^2 downward.
Projectile
An object moving under gravity alone in two dimensions after launch.
Reference frame
The observer's coordinate system relative to which positions and velocities are measured.
Relative velocity
The velocity of one object as measured in the reference frame of another.
Exam technique
Always choose a positive direction at the start and apply the sign consistently to every vector quantity in the problem.
Do not assume the sign of acceleration tells you speeding up versus slowing down; compare the signs of velocity and acceleration instead.
Remember that an object at the top of its flight has zero velocity but still has acceleration g downward.
On a velocity versus time graph, slope is acceleration and area under the line is displacement; mixing these up is a common trap.
For projectiles, treat horizontal and vertical motion separately but use the same time t to link them.
List your knowns and the target unknown, then pick the kinematic equation that omits the variable you neither have nor need.
Quick check
A ball is thrown straight up into the air. At the highest point of its path, what are its velocity and acceleration?
Velocity is zero and acceleration is zero
Velocity is zero and acceleration is about 9.8 m/s^2 downward
Velocity is maximum and acceleration is zero
Velocity is about 9.8 m/s upward and acceleration is zero
Show answer
Answer: VELOCITY IS ZERO AND ACCELERATION IS ABOUT 9.8 M/S^2 DOWNWARD. At the top of its flight the ball is momentarily at rest, so its velocity is zero. Gravity never stops acting, however, so the acceleration remains g, about 9.8 m/s^2 directed downward, which is exactly what pulls the ball back down.