Uniform Linear Motion

Uniform linear motion (ULM) is a motion in which an object moves in a straight line at constant velocity, and therefore with zero acceleration. Its position evolves linearly with time. It is only defined in a given reference frame, which must always be specified.

Discover FizziQ

How to measure it in class

Two routes with FizziQ: video tracking, which should give a straight line on x(t), and the accelerometer, which should stay at zero.

Steps:

Place the smartphone on a cart or a sliding object, and give it a push on a level, smooth surface.
Record the linear acceleration (without g) during the motion: after the initial push, the value should oscillate around zero as long as friction remains low.
Film the same experiment with a fixed camera, then track the moving object frame by frame in the Kinematics module after calibrating the scale.
Plot x(t): points lining up is the signature of ULM. Model with a straight line and read off the slope, which is the velocity.
Plot v(t): the curve should be horizontal. Compare its value with the one obtained from the slope of x(t).
Repeat on a rough surface: the decrease of v(t) shows that friction, and not the absence of a push, is what slows the object down.

Scientific activities on this topic

Using a smartphone’s accelerometer or video camera, you can carry out many experiments on uniform linear motion:

Trajectory of a football shot
Steering a rover on Mars
Inertial reference frame
Falcon 9 landing kinematics - identify, in the launcher’s descent, the phase at nearly constant velocity before the final engine burn.

Learn more

The principle of inertia, Newton’s first law

A body on which the forces balance, or on which no force acts, keeps its uniform linear motion, or remains at rest. In other words: no force is needed to maintain a uniform linear motion. A force is only required to change it. This is the point students find hardest to accept, because everyday experience says the opposite.

The break with Aristotle

For Aristotle, a moving object stops as soon as you stop pushing it: motion would require a permanent cause. The intuition is correct in the real world, but the cause of the stopping is friction, not the absence of a push. Galileo and then Newton understood that with friction removed, the object would never stop. On an air cushion table or in the vacuum of space, a launched object keeps going indefinitely.

Rest and ULM are the same state

A motionless object has a constant velocity, equal to zero: rest is a special case of ULM. Better still, no mechanics experiment can distinguish a reference frame at rest from a reference frame in ULM relative to the first. This is the principle of Galilean relativity, and it is why you feel nothing in a train running at constant speed in a straight line. The inertial reference frame entry develops this point.

Frequent errors

Believing that “constant speed” is enough to characterize ULM: the velocity vector must be constant, and therefore also its direction. A uniform circular motion has a constant speed magnitude and is not a ULM. Another confusion: thinking that zero acceleration means zero velocity. A TGV at 300 km/h on level track has zero acceleration.

Orders of magnitude

Walking: about 1.4 m/s. Cruising bicycle: 5 m/s. Car on a highway: 36 m/s (130 km/h). TGV: 83 m/s. Cruising airliner: 250 m/s. The Voyager 1 probe, in nearly uniform linear motion outside the solar system: about 17,000 m/s.

Formula

The position evolves linearly with time:

x(t) = x₀ + v·t

The velocity is constant:

v(t) = v = constant

The acceleration is zero:

a(t) = 0

The velocity is calculated from the distance traveled:

v = d / Δt

where:

x(t): position at time t (m)
x₀: position at time t = 0 (m)
v: velocity, constant in magnitude, direction and sense (m/s)
t: time (s)
d: distance traveled (m)
Δt: duration of the journey (s)
a: acceleration (m/s²)

Application examples

A car on cruise control on a straight highway at 130 km/h covers 36 m every second; in 10 s, it advances 360 m.
A curling stone glides across the ice in a straight line at almost constant speed, because friction there is very low.
A spacecraft with its engines off, far from any celestial body, continues indefinitely on its straight trajectory at constant speed.
Sound travels in a straight line at 340 m/s in homogeneous air: an echo returning after 2.0 s indicates an obstacle 340 m away.
An airport moving walkway of 60 m traveled in 50 s corresponds to a speed of 1.2 m/s.
A marble dropped into a tube of glycerin reaches its terminal velocity, then descends in uniform linear motion: the weight is then exactly balanced by the buoyant force and the friction.

FAQ

Q: Is a force needed for an object to move at constant velocity? A: No. It is the opposite of intuition, but it is what Newton’s first law says. If the forces balance, the uniform linear motion maintains itself. A car burns fuel not to move forward, but to compensate for friction and air resistance.

Q: Is rest a uniform linear motion? A: Yes, it is the special case where the velocity is zero. Nothing in physics distinguishes an object at rest from an object in ULM: everything depends on the reference frame from which you observe it.

Q: Is a circular motion at constant speed a ULM? A: No. The magnitude of the velocity is constant, but its direction changes constantly, so the velocity vector varies and the acceleration is not zero. The word “linear” is essential.

Q: Why does my accelerometer never read exactly zero during a ULM? A: The sensor picks up the vibrations of the support, the small irregularities of the surface and its own electronic noise. The motion is considered uniform if the values oscillate around zero without systematic drift.

Q: Does a perfect ULM exist in reality? A: Almost never on Earth, because of friction and gravity. It is a model: you use it when the forces balance well enough, for example on an air cushion table, on ice, or in interstellar space.

Inertial Reference Frame - Kinematics - Uniformly Accelerated Linear Motion - Uniform Circular Motion - Friction Force