Kinematics

Kinematics describes the motion of an object using geometric quantities - position, velocity and acceleration - without considering the forces that cause them, which is the subject of dynamics. Any description only makes sense with respect to a chosen and clearly stated reference frame.

Discover FizziQ

How to measure it in class

FizziQ’s Kinematics module turns a video or a chronophotograph into a table of positions, then into velocity and acceleration curves.

Steps:

Film the motion with a fixed camera, in a plane perpendicular to the displacement, with an object of known length (ruler, tape measure) visible in the frame for calibration.
Open FizziQ’s Kinematics module and import the video, or choose a video from the built-in library.
Define the coordinate system: place the origin of the axes and calibrate the scale by clicking the two ends of the object of known length.
Track the same point of the moving object frame by frame; the app automatically calculates the times and the x and y positions.
Plot x(t) and y(t), then display the velocities. A straight line on v(t) indicates constant acceleration, a horizontal line indicates uniform motion.
Export the data to the spreadsheet or the experiment notebook for modeling and the lab report.

Scientific activities on this topic

Using the videos and chronophotographs in the sports video library, carry out kinematic analyses of athletes’ movements:

Other activities deal with measuring position, velocity and acceleration:

Speed by magnetic markers: measure a speed by timing the passage past magnetic markers.
Measuring g by video analysis: work back to the acceleration of gravity from video tracking.
Takeoff speed of an airplane: integrate the measured acceleration to obtain the rotation speed.
Cycloid of a rolling wheel: show that a point on a wheel traces a cycloid in the ground reference frame.
Straight-line course without a compass: hold a straight course using only the gyroscope.
Center of gravity of a diver: identify the point whose trajectory remains parabolic during the figures.
Linear acceleration: principle: understand what an accelerometer really measures.
Falcon 9 landing kinematics: analyze the braking phase of a reusable launcher.
Elevator speed by barometer: deduce a vertical speed from the pressure variation.

The kinematics video and chronophotography library contains many other videos to use in class or at home (open the tool).

Learn more

The three quantities of kinematics

Position locates the moving object in a coordinate system attached to the reference frame. Velocity is the derivative of position with respect to time, acceleration the derivative of velocity. These three quantities are vectors: they have a magnitude, a direction and a sense. Confusing the magnitude with the vector is the source of most errors, in particular in curved motions where the magnitude can remain constant while the vector changes.

The reference frame comes first

No trajectory exists “in itself”. A point on a bicycle wheel traces a circle in the frame’s reference frame and a cycloid in the ground reference frame. A seated passenger is motionless in the train’s reference frame and in uniform linear motion in the platform’s. Before characterizing a motion, you therefore always state the reference frame of study; the inertial reference frame entry specifies which reference frames allow Newton’s laws to be applied.

The reference motions of the curriculum

Three cases are studied in high school: uniform linear motion (constant velocity vector, zero acceleration), uniformly accelerated linear motion (constant acceleration, straight trajectory) and uniform circular motion (constant speed magnitude, non-zero centripetal acceleration). Each has its own dedicated entry.

Decomposing a complex motion

A projectile motion is neither linear nor circular, but it reduces to two simple motions projected onto the axes: a uniform linear motion horizontally and a uniformly accelerated linear motion vertically with acceleration g. This is the idea introduced by Galileo: treat the components of a single vector separately.

Orders of magnitude

Walking: about 1.4 m/s. 100 m sprinter: up to 12 m/s. Kicked football: 25 to 35 m/s. Car on a highway: 36 m/s. Free fall, acceleration: 9.81 m/s². Emergency braking of a car: 6 to 9 m/s². Takeoff of an airliner: about 2 to 3 m/s².

FizziQ is the ideal tool for studying kinematics. The app’s kinematics module can analyze both videos and chronophotographs, it is easy to use, gives precise results that students can quickly work with, and the data can be shared in a few clicks.

You can also discover all the advanced features, such as data management in a table or graph creation, in our complete guide to video analysis and chronophotography.

Note that FizziQ is available on smartphones and tablets, but also on computers if you want to run analyses with a large screen and a mouse (read the article).

Formula

The average velocity between two times is the ratio of the displacement to the duration:

v_avg = Δx / Δt

The instantaneous velocity is the derivative of position with respect to time:

v(t) = dx/dt

The acceleration is the derivative of velocity with respect to time:

a(t) = dv/dt = d²x/dt²

In video tracking, the velocity at point i is approximated by the method of surrounding points:

v_i = (x_(i+1) − x_(i−1)) / (t_(i+1) − t_(i−1))

where:

x: position of the moving object along the axis of study (m)
t: time (s)
Δx: change in position between two times (m)
Δt: elapsed time (s)
v: velocity (m/s)
a: acceleration (m/s²)

Application examples

A sprinter covers 100 m in 10.0 s: his average speed is 10.0 m/s, while his instantaneous speed exceeds 12 m/s in the middle of the race.
Video tracking of a basketball gives a vertical acceleration of 9.8 m/s² throughout the flight, confirming that only gravity is acting.
The kinematic analysis of a pole vault makes it possible to follow the athlete’s center of gravity and calculate the maximum height reached.
In robotics, calculating the position of a gripper from the joint angles is a problem of pure kinematics, with no force involved.
Studying the landing of a Falcon 9 launcher on video shows a deceleration of several tens of m/s² during the final engine burn.
A smartphone GPS gives a position every second; differentiating these positions provides the speed displayed by running apps.

FAQ

Q: What is the difference between kinematics and dynamics? A: Kinematics describes the motion (position, velocity, acceleration) without asking why it occurs. Dynamics relates this motion to forces, through Newton’s laws. Kinematics always comes first.

Q: Why must the reference frame be specified? A: Because the trajectory, velocity and acceleration depend on the chosen reference frame. A train passenger is motionless for their neighbor and moving at 300 km/h for an observer on the platform. Without a stated reference frame, a kinematic answer is meaningless.

Q: What frame rate is needed for video tracking? A: 25 to 30 frames per second are enough for a projectile motion or a human movement. For a collision or a very fast fall, switch to the smartphone’s slow-motion mode, at 120 or 240 frames per second.

Q: Why is my velocity curve noisy while the position curve is smooth? A: Differentiation amplifies errors. A tracking imprecision of a few pixels becomes a significant error on the velocity, and even more on the acceleration. You must carefully track the same point of the moving object in each frame and calibrate the scale correctly.

Q: Can kinematics be done without video? A: Yes. The smartphone’s accelerometer directly gives the acceleration, which can be integrated to obtain the velocity. The barometer gives the altitude, the GPS the position. Each sensor opens a different route to the same quantities.

Inertial Reference Frame - Uniform Linear Motion - Uniformly Accelerated Linear Motion - Uniform Circular Motion - Chronophotography - Accelerometer