Kinematics of a particle
Kinematics describes motion without asking about its causes. It consists of three functions of time —position, velocity and acceleration— linked by a derivative, and that same structure reappears later in the charging of a capacitor.
01Describing a motion
Before asking why something moves, you have to be able to say how it moves. That takes three prior decisions, which are almost always made without noticing:
- A frame of reference. There is no absolute motion: a seated passenger is at rest relative to the bus and moving at 60 km/h relative to the street. Both descriptions are correct.
- A time origin. The instant called t = 0.
- A model of the body. If the size of the object does not matter for the problem, it is treated as a particle (point mass): all its mass concentrated at a point. A satellite in orbit is a point; the same satellite pointing its antenna is not.
The position r⃗(t) is the primary datum. Velocity is its derivative and acceleration is the derivative of velocity:
And the other way around, by integrating: from the acceleration you recover the velocity and from that the position, at the cost of two constants, which are the initial velocity and the initial position.
This structure —a quantity, its derivative and its second derivative— is the same one found in the differential equations that describe the charging of a capacitor. It is not a forced analogy: it is literally the same mathematics with different names.
02Rectilinear motion
| Case | Acceleration | Velocity | Position |
|---|---|---|---|
| Uniform (constant velocity) | 0 | v constant | \( x = x_{0} + v\cdot t \) |
| Uniformly accelerated (constant acceleration) | a constant | \( v = v_{0} + a\cdot t \) | \( x = x_{0} + v_{0}\cdot t + \tfrac{1}{2}\cdot a\cdot t^{2} \) |
| Free fall | \( g \approx 9.8 \ \mathrm{m/s^2} \) downward | \( v = v_{0} - g\cdot t \) | \( y = y_{0} + v_{0}\cdot t - \tfrac{1}{2}\cdot g\cdot t^{2} \) |
Eliminating time between the last two columns of the constant-acceleration case gives a very useful relation, because it does not require knowing how long the motion lasted:
Average velocity is displacement divided by time, and it can be zero even if the body has moved a great deal: it only has to return to its starting point. Average speed, on the other hand, uses the distance traveled and is never negative. On an x–t graph, the average velocity is the slope of the secant line and the instantaneous velocity is the slope of the tangent.
Choose the type of motion and move through time with the slider. The numbers on the right come from differentiating the position, not from a table.
03Reading graphs
A good share of kinematics problems are solved by looking rather than calculating, if you remember two rules:
The slope of a graph is the derivative and the area under it is the integral.
- Slope of x–t = velocity. Slope of v–t = acceleration.
- Area under v–t = displacement. Area under a–t = change in velocity.
That is why, in constant-acceleration motion, the area under v–t is a trapezoid, and its formula —average base times height— is exactly x = (v₀ + v)/2 · t.
The area under the curve counts with sign: a stretch of v–t below the axis subtracts displacement. It is the same distinction as between the definite integral and geometric area.
04Motion in the plane: projectile motion
The principle of independence of motions says that the x and y components can be treated separately. In projectile motion, with air resistance neglected:
| Axis | Type of motion | Equations |
|---|---|---|
| Horizontal | Constant velocity: there is no horizontal force | \( x = v_{0}\cdot \cos\theta \cdot t \) |
| Vertical | Constant acceleration with \( a = -g \) | \( y = v_{0}\cdot \operatorname{sin}\theta \cdot t - \tfrac{1}{2}\cdot g\cdot t^{2} \) |
From these come the three results worth memorizing, for a launch from the ground:
05Circular motion
In uniform circular motion the magnitude of the velocity does not change, but its direction does: there is acceleration, and it always points toward the center.
The angular velocity ω in rad/s is the same ω as the angular frequency of a sinusoidal signal. Projecting onto an axis a point that rotates with constant angular velocity gives exactly a sinusoid: that is the origin of the phasor used to solve AC circuits, and the reason an alternator generates a sinusoidal voltage as it rotates.
06Relative motion
If a body moves relative to a frame that is itself moving, the velocities add as vectors: v⃗AC = v⃗AB + v⃗BC. This explains why a swimmer who crosses a river perpendicular to the bank ends up downstream, and why you have to aim upstream to reach the point directly opposite.
The same reasoning, applied to waves instead of bodies, produces the Doppler effect, and when the velocities approach that of light, simple addition no longer holds: that is where relativity begins.
07In the lab
Film an object falling at 240 frames per second against a vertical ruler. Measure the position every five frames, plot y–t and fit a parabola. The quadratic coefficient is −g/2.
With a water jet at constant flow rate, measure the range for five angles between 15° and 75°. Verify that the maximum is at 45° and that complementary angles give equal ranges.
Build two gates with an LED and a phototransistor 20 cm apart and use a microcontroller to measure the time between interrupts. It is the same principle as a motor encoder, and it lets you measure the instantaneous velocity of a cart on an inclined plane.
08Common mistakes
- Confusing velocity with speed, or displacement with distance traveled.
- Believing that zero acceleration implies zero velocity. A body can move fast without accelerating, and be instantaneously at rest with maximum acceleration: that is what happens at the highest point of a vertical throw.
- Using the constant-acceleration formulas when the acceleration is not constant.
- Mixing up the axes in projectile motion and applying gravity to the horizontal motion.
- Assuming there is no acceleration in uniform circular motion because the speed does not change. The direction does change, and that is already acceleration.
- Forgetting the signs: once the positive direction of the axis is fixed, you have to stick to it to the end.
09Self-assessment
A car goes from 0 to 100 km/h in 8 s. What is its average acceleration?
100 km/h = 27.8 m/s, so a = 27.8/8 ≈ 3.5 m/s², a bit more than a third of g.
A stone is thrown upward at 20 m/s. How high does it go and how long does it take to come back?
h = v₀²/(2g) = 400/19.6 ≈ 20.4 m; it takes v₀/g ≈ 2.04 s to rise and twice that, 4.08 s, to return to the starting point.
What does the area under a v–t graph represent?
The displacement, counted with sign.
Two projectiles leave at 30 m/s, one at 30° and the other at 60°. Which one lands farther away?
Neither: the angles are complementary and the range is the same, about 79.5 m. The one at 60° goes higher and takes longer to land.
A point rotates at 3000 rpm at a radius of 4 cm. What is its centripetal acceleration?
ω = 3000·2π/60 = 314 rad/s, so a = ω²r = 314²·0.04 ≈ 3950 m/s², about 400 times g. That is why rotors are balanced so carefully.
Why does a uniformly rotating point generate a sinusoid when projected?
Because its coordinate is x = r·cos(ωt): the same expression as an AC signal of amplitude r and angular frequency ω. This is where the phasor representation comes from.
10Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 2 to 4: the standard text for the course, with graded problems.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). More discursive explanations and good examples of relative motion.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume I: Mainly Mechanics, Radiation, and Heat. Addison-Wesley, 1963 (Spanish edition, “Física, volumen I: mecánica, radiación y calor”, Addison-Wesley / Pearson). Chapters 8 and 9 show where the differential calculus applied to motion comes from.
- Marcelo Alonso and Edward J. Finn. Physics. Addison-Wesley, 1992 (Spanish edition, “Física”, Addison-Wesley Iberoamericana, 1995). The classic of Argentine engineering schools, with a vector treatment from the start.