Kinematics in one and two dimensions

Kinematics is the description of motion: where something is, how fast it is going and how quickly that is changing, without yet asking why. It is the language that the rest of mechanics is written in. Stopping distances on highways, the trajectory of water from a fire hose, the timing of a conveyor system and the path a drone must fly to hold position in a crosswind are all kinematics problems.

This chapter defines position, velocity and acceleration precisely, connects them through calculus and graphs, develops the constant-acceleration equations and applies them to braking, free fall and projectiles. It finishes with relative velocity, which you need whenever motion is measured from something that is itself moving.

Position, displacement and distance

To describe motion along a straight line, choose an origin and a positive direction. The position \(x\) is the signed distance from the origin. Displacement is the change in position, \(\Delta x = x_2 - x_1\). It can be negative, and it depends only on where you start and finish.

Distance traveled is different: it is the total length of path covered, always positive. A delivery van that drives 8 km east and then 3 km west has traveled 11 km but has a displacement of only 5 km east. Engineers need both. Fuel use and tire wear depend on distance; where the van ends up depends on displacement.

Velocity and speed

Average velocity is displacement divided by elapsed time, \(\bar{v} = \Delta x/\Delta t\). Average speed is distance divided by time. They differ whenever the motion reverses direction.

Instantaneous velocity is the limit of the average velocity over a shorter and shorter interval, which is exactly the derivative from Calculus I:

On a graph of position against time, the velocity at any instant is the slope of the tangent line. A steep upward slope means fast motion in the positive direction; a flat graph means the object is momentarily at rest. Speed is the magnitude of velocity.

Acceleration

Acceleration is the rate of change of velocity, the slope of the velocity–time graph:

Its unit is meters per second per second, m/s². Acceleration has a sign, and the sign is about direction, not about "speeding up". A car moving in the positive direction and braking has negative acceleration. A car moving in the negative direction and braking has positive acceleration. An object speeds up when its velocity and acceleration have the same sign, and slows down when they have opposite signs.

Running the calculus backward, velocity is the integral of acceleration and position is the integral of velocity. Graphically, the area under the acceleration–time graph is the change in velocity, and the area under the velocity–time graph is the displacement.

Motion with constant acceleration

Many practical situations have acceleration that is constant, or close enough: a vehicle braking hard, an object falling over short distances, a load on a steadily accelerating hoist. Integrating \(a\) = constant gives four equations that relate the five quantities \(x - x_0\), \(v_0\), \(v\), \(a\) and \(t\):

Each equation leaves out one quantity. The working method: list the three quantities you know and the one you want, then pick the equation that contains exactly those four. The third equation, which has no \(t\), is the one to use for stopping distances.

Because braking distance grows with \(v^2\), doubling speed quadruples it. At 120 km/h the braking distance in the example becomes 79 m, not 66 m. This is the physics behind speed limits in work zones and school areas.

Free fall

Near the Earth's surface, and when air resistance is small, every object falls with the same constant acceleration \(g = 9.81\) m/s², directed downward. With upward taken as positive, \(a = -g\). The constant-acceleration equations apply directly.

Notice that the ball passes the roof on the way down at 15 m/s, the same speed it was thrown up with. Without air resistance, motion under gravity is symmetric in time.

Motion in two dimensions

In a plane, position, velocity and acceleration are vectors: \(\mathbf{r} = (x, y)\), \(\mathbf{v} = d\mathbf{r}/dt\), \(\mathbf{a} = d\mathbf{v}/dt\). The key simplification is that the \(x\) and \(y\) motions are independent. Each component obeys its own one-dimensional equations, linked only by the shared time \(t\).

Projectile motion

A projectile moves under gravity alone. Its horizontal acceleration is zero and its vertical acceleration is \(-g\). So the horizontal motion is uniform and the vertical motion is free fall:

Eliminating \(t\) shows that the path is a parabola. For a launch and landing at the same height, three useful results follow:

for the time of flight, maximum height and range. The range is greatest at \(\theta = 45^\circ\), and complementary angles (such as 30° and 60°) give the same range.

The level-ground formulas do not apply when launch and landing heights differ. Then go back to the component equations.

Relative velocity

Velocity is always measured relative to something. If a boat moves at \(\mathbf{v}_{BW}\) relative to the water and the water moves at \(\mathbf{v}_{WG}\) relative to the ground, the boat's velocity relative to the ground is the vector sum

The subscripts chain like dominoes: the inner letters match and cancel. The same rule handles aircraft in wind, rain seen from a moving car and parts on a moving conveyor.