Uniformly accelerated motion: smartphone experiments

Uniformly accelerated linear motion is the motion of a moving object in a straight line whose acceleration is constant. The velocity then changes by the same amount every second, while the position evolves as the square of time. Free fall is the typical case.

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How to measure it in class

FizziQ makes it possible to verify the two signatures of uniformly accelerated linear motion: a constant acceleration on the accelerometer, and a parabola on x(t) in video tracking.

Steps:

Film, with a fixed camera, the fall of a ball along a wall, with a tape measure visible for calibration, in slow-motion mode if the smartphone allows it.
Import the video into the Kinematics module, calibrate the scale and mark the ball frame by frame.
Plot y(t): the points should line up on a parabola. Model it with a degree-2 polynomial and read the coefficient of t², which equals a/2.
Plot v(t): the curve should be a straight line. Its slope directly gives the acceleration, to be compared with 9.81 m/s².
Repeat the measurement on an inclined plane: the acceleration is then g·sin(α), smaller, and therefore easier to track than a vertical fall.
Place the smartphone in an elevator and record the linear acceleration: identify the start-up phase at constant acceleration, then the constant-speed phase.

Scientific activities on this topic

Use a phone’s accelerometer or a smartphone’s kinematic analysis to understand what uniformly accelerated linear motion is:

How can the Opportunity rover on Mars move in a straight line?
Study of a free fall to calculate gravity
Analysis of the trajectory of a basketball
Galileo’s inclined plane - rediscover the historical law of distances proportional to the square of time.
Takeoff speed of an airplane - measure the acceleration during the takeoff roll and deduce the rotation speed.

Learn more

Galileo’s law of distances

Around 1604, Galileo rolled balls down inclined planes and found that the distance traveled is proportional to the square of time. The distances traveled during successive equal time intervals are in the ratios 1, 3, 5, 7… This is the first quantitative law of falling bodies, and it contradicts Aristotle, for whom the speed of fall depended on weight. The inclined plane serves to “dilute” gravity to make it measurable with the clocks of the time.

Zero velocity does not imply zero acceleration

This is the most frequent confusion. Throw a ball vertically: at the top of its trajectory, its velocity is zero for an instant, but its acceleration is still 9.81 m/s² downward. If the acceleration were zero there, the ball would remain suspended in the air. Velocity and acceleration are two independent quantities: one says where you are going, the other how that is changing.

Negative acceleration, deceleration, increasing speed

The sign of a depends on the orientation of the chosen axis, not on whether the object is speeding up or slowing down. The object slows down when v and a have opposite signs, and speeds up when they have the same sign. A negative acceleration with a negative velocity corresponds to an object moving faster and faster in the negative direction. So always reason on the relative signs, never on the sign of a alone.

Decomposing projectile motion

A ball thrown into the air is not in uniformly accelerated linear motion, since its trajectory is curved. But by projecting onto two axes, one obtains a uniform linear motion horizontally (a_x = 0, neglecting air) and a uniformly accelerated linear motion vertically (a_y = −g). The two components are independent: this is why a ball dropped and a ball fired horizontally from the same height hit the ground at the same time.

A motion always defined relative to a reference frame

Like any motion, uniformly accelerated linear motion only makes sense relative to a reference frame. It is in the Earth’s reference frame that a vertical free fall is a uniformly accelerated linear motion. The term “linear” (rectilinear) refers to motion along a single straight line: as soon as a motion takes place along several axes, it is decomposed axis by axis for analysis.

Orders of magnitude

Free fall: 9.81 m/s². Inclined plane at 10°: 1.7 m/s². City car from 0 to 100 km/h in 10 s: 2.8 m/s². Emergency braking on dry ground: 6 to 9 m/s². Takeoff of an airliner: 2 to 3 m/s². Rocket at liftoff: 15 to 30 m/s². Airbag during a crash: several hundred m/s².

Formula

The position varies as the square of time:

x(t) = x₀ + v₀·t + ½·a·t²

The velocity varies linearly with time:

v(t) = v₀ + a·t

The acceleration is constant:

a(t) = a = constant

A time-free relation links velocity and distance traveled:

v² − v₀² = 2·a·(x − x₀)

where:

x(t): position at time t (m)
x₀: initial position (m)
v(t): velocity at time t (m/s)
v₀: initial velocity (m/s)
a: acceleration, constant (m/s²)
t: time (s)

Application examples

A ball dropped without initial velocity falls ½ × 9.81 × 1² ≈ 4.9 m in 1 s, and 19.6 m in 2 s: four times as far for twice the time.
A car going from 0 to 100 km/h (27.8 m/s) in 10 s has an average acceleration of 2.8 m/s² and covers about 139 m.
A ball thrown upward at 15 m/s reaches its maximum height after 1.53 s, at 11.5 m: at that instant its velocity is zero but its acceleration is still 9.81 m/s².
An airliner accelerates at about 2.5 m/s² on the runway and reaches 70 m/s (250 km/h) after about 28 s and 1,000 m of ground roll.
A skier on a 15° slope experiences an acceleration of g·sin(15°) ≈ 2.5 m/s², reduced by the friction of the snow.
The first stage of a launcher sees its acceleration increase during the flight as it empties of propellant: the motion is therefore not exactly uniformly accelerated.

FAQ

Q: Can an object have zero velocity and non-zero acceleration? A: Yes, and this is the case at the top of a vertical throw. The velocity vanishes for an instant as it changes sign, but gravity acts without interruption: the acceleration is 9.81 m/s² downward throughout the flight, including the ascent.

Q: Does negative acceleration mean the object is slowing down? A: Not necessarily. It depends on the direction of the velocity. The object slows down if v and a have opposite signs. If both are negative, it moves faster and faster in the negative direction of the axis.

Q: Why use an inclined plane rather than a vertical fall? A: Because the acceleration there equals g·sin(α), and is therefore much smaller. The motion is slower, easier to film and to track, and the measurement uncertainties decrease. This is the trick Galileo was already using.

Q: Is projectile motion a uniformly accelerated linear motion? A: No, its trajectory is parabolic, and therefore not linear. But each of its two components is a simple case of it: uniform linear motion horizontally, uniformly accelerated linear motion vertically.

Q: Why is my measured value of g lower than 9.81 m/s²? A: Air resistance slows the object, especially if it is light or bulky. A calibration error in the video tracking scale also acts directly on the result. Use a dense, compact object, and check the calibration.

Linear Acceleration - Free Fall - Kinematics - Uniform Linear Motion - Inertial Reference Frame - Inclined Plane - Parabolic Trajectory (Projectile) - Accelerometer

Uniformly Accelerated Linear Motion