Working with a stroboscopic picture

The complete method, from reading the motion to the acceleration vector, with a worked example and mistakes to avoid.

What is a stroboscopic picture?

It is a picture showing an object's successive positions, recorded at equal time intervals τ (for example every 40 ms). It can be made with a stroboscope, a long-exposure photograph or, today, by tracking a video frame by frame. The positions are labelled M₁, M₂, M₃…

Stroboscopic picture of a thrown ball (τ = 60 ms); the length scale is at the bottom left.

1. Reading the motion without calculation

What you see	What it means
Points in a line	Straight-line motion
Points on a circle	Circular motion
Equal gaps	Constant speed: uniform motion
Growing gaps	Speed increasing: accelerating
Shrinking gaps	Speed decreasing: slowing down

2. Computing a speed

The speed at point M_i is computed from the two points on either side:

$v_i = \dfrac{M_{i-1}M_{i+1}}{t_{i+1} - t_{i-1}} = \dfrac{M_{i-1}M_{i+1}}{2\tau}$

Measure M_i−1M_i+1 on the picture with a ruler, then convert it into a real distance using the scale. Example: in the figure, M₃M₅ stands for 45.8 cm; with τ = 60 ms, $v_4 = 0.458 / 0.12 = 3.82$ m/s.

Why the neighbouring points rather than M_iM_i+1? Because the mean speed between M_i−1 and M_i+1 is a good estimate of the speed at the middle time t_i. As a result, there is no speed at the first and last points.

3. Drawing the velocity vector

Starting point: the point M_i.
Direction: parallel to the line (M_i−1M_i+1), i.e. tangent to the trajectory.
Sense: the direction of motion.
Length: proportional to v_i, using a chosen velocity scale (for example 1 cm for 0.5 m/s), shown on the figure.

In red, the velocity vectors at M₄ and M₆; in grey, the same vectors redrawn from M₅; in blue, the change-in-velocity vector Δv₅.

4. Change in velocity and acceleration

Redraw v_i−1 and v_i+1 from point M_i; the vector from the tip of v_i−1 to the tip of v_i+1 is the change-in-velocity vector:

$\Delta\vec v_i = \vec v_{i+1} - \vec v_{i-1}\qquad\text{and}\qquad \vec a_i \approx \dfrac{\Delta\vec v_i}{t_{i+1} - t_{i-1}} = \dfrac{\Delta\vec v_i}{2\tau}$

In the example, Δv₅ is vertical and points down, and a₅ ≈ 9.8 m/s²: this is the acceleration due to gravity.

5. Linking to forces

Newton's first law: if the velocity vector is constant (uniform straight-line motion), the forces balance, and conversely.
The net force points the same way as Δv.
Newton's second law: $\sum \vec F = m\,\vec a$. For free fall or a throw without air resistance, only the weight acts, so $\vec a = \vec g$.

Common mistakes

Forgetting the scale: the distance measured on paper is not the real distance.
Dividing by τ instead of 2τ when using M_i−1M_i+1.
Drawing the velocity vector towards the next point M_i+1 instead of parallel to (M_i−1M_i+1).
Mixing up the length scale and the velocity scale.
Concluding that circular motion “at constant speed” has no acceleration: the direction of the velocity changes, so Δv is not zero.

Practice

Five printable worked exercises apply this method (uniform motion, falling, throwing, circular motion, braking). To make your own stroboscopic pictures from a video, use the video tracking tool: it also computes and draws the vectors so you can check your constructions.