Vectors and motion with changing acceleration

Lesson 1 described motion along a line and in a plane, mostly with constant acceleration. Real motion is rarely that tidy. A lift accelerates smoothly from rest, a boat coasts to a stop as water drag fades with its speed, a truck slows in an arrester bed that resists harder the deeper it goes, and a centrifuge spins its samples at thousands of times the acceleration of gravity. This lesson builds the tools for these cases. It begins with vectors, the language for quantities with direction, including the dot and cross products that later lessons use for work and torque. It then treats motion in which the acceleration depends on time, on velocity or on position, using calculus; treats circular motion with tangential and centripetal acceleration; and ends with step-by-step numerical solution, the method engineers use when no formula exists.

Vectors and components

Displacement, velocity, acceleration and force have both size and direction: they are vectors. In a coordinate system with unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) along x, y and z, a vector is written by its components, \(\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}\), with magnitude \(|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}\). In two dimensions, a vector of magnitude A at angle θ from the x-axis has components \(A_x = A\cos\theta\) and \(A_y = A\sin\theta\), and going back, \(\theta = \arctan(A_y/A_x)\), with care about the quadrant. Vectors add component by component, which is far more reliable than adding them graphically.

The dot and cross products

Two ways of multiplying vectors appear throughout mechanics. The dot (scalar) product gives a number:

It measures how much two vectors point the same way: it is zero for perpendicular vectors. Lesson 5 uses it to define work, \(W = \mathbf{F} \cdot \mathbf{d}\). The cross (vector) product gives a vector perpendicular to both, with magnitude \(AB\sin\theta\) and direction set by the right-hand rule:

Lesson 8 uses it for torque, \(\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\), and angular momentum. Note that \(\mathbf{B} \times \mathbf{A} = -\mathbf{A} \times \mathbf{B}\): order matters.

When acceleration depends on time

If the acceleration is a known function of time, velocity and position follow by integration (Lesson 1, MATH-101): \(v(t) = v_0 + \int_0^t a\,dt\) and \(x(t) = x_0 + \int_0^t v\,dt\). Smooth acceleration profiles are common in engineering because sudden changes in acceleration (jerk) are uncomfortable for people and hard on machinery.

When acceleration depends on velocity

Drag forces depend on speed, so the acceleration is a function of v. Separate the variables in \(dv/dt = a(v)\) and integrate (MATH-201). For drag proportional to speed, \(a = -kv\):

When acceleration depends on position

Springs, arrester beds and gravitational fields give accelerations that depend on position. The chain rule rewrites the acceleration as \(a = \dfrac{dv}{dt} = \dfrac{dv}{dx}\dfrac{dx}{dt} = v\dfrac{dv}{dx}\), so \(v\,dv = a(x)\,dx\), which can be integrated directly to relate speed and position. This is a first look at the energy method of Lesson 5.

Circular motion

An object moving on a circle of radius r with speed v is accelerating even if its speed is constant, because its velocity keeps changing direction. The acceleration has two perpendicular parts: a centripetal part pointing to the center and a tangential part along the path that changes the speed (MATH-209):

where \(\omega = v/r\) is the angular speed in radians per second (Lesson 8). The period of one revolution is \(T = 2\pi r/v = 2\pi/\omega\).

Numerical solution

Many motions have no formula: a projectile with drag proportional to v², a vehicle with a complicated engine map, a satellite perturbed by the Moon. They are solved step by step. Starting from the known position and velocity, compute the acceleration, advance the velocity and position over a short time step Δt as if the acceleration were constant during it, and repeat. This is Euler's method (ENCM-100, MATH-300). Its error shrinks in proportion to Δt, so halving the step until the answer stops changing gives confidence in the result.

import math

# Ball launched at 50 m/s, 30 degrees; drag acceleration k v^2 opposite the velocity
g, k, v0, angle = 9.81, 0.004, 50.0, math.radians(30)

def flight(k, dt):
    x, y = 0.0, 0.0
    vx, vy = v0 * math.cos(angle), v0 * math.sin(angle)
    t, ymax = 0.0, 0.0
    while y >= 0.0:
        v = math.hypot(vx, vy)
        ax, ay = -k * v * vx, -g - k * v * vy
        x, y = x + vx * dt, y + vy * dt           # step position with the old velocity (Euler)
        vx, vy = vx + ax * dt, vy + ay * dt       # then update velocity
        t += dt
        ymax = max(ymax, y)
    return x, ymax, t

print(f"Vacuum formula: range {v0**2 * math.sin(2 * angle) / g:.1f} m")
for dt in [0.1, 0.01, 0.001]:
    R, H, T = flight(0.0, dt)
    print(f"No drag, dt = {dt}: range {R:6.1f} m")
for dt in [0.1, 0.01, 0.001]:
    R, H, T = flight(k, dt)
    print(f"With drag, dt = {dt}: range {R:6.1f} m, height {H:5.1f} m, time {T:4.2f} s")

Vacuum formula: range 220.7 m
No drag, dt = 0.1: range  225.2 m
No drag, dt = 0.01: range  221.3 m
No drag, dt = 0.001: range  220.7 m
With drag, dt = 0.1: range  144.6 m, height  25.7 m, time 4.60 s
With drag, dt = 0.01: range  141.7 m, height  24.8 m, time 4.49 s
With drag, dt = 0.001: range  141.3 m, height  24.8 m, time 4.48 s

Without drag, the numerical range converges to the formula value of 220.7 m as the step shrinks, which checks the program. With drag, there is no formula, and the converged answer, about 141 m, is a third shorter, with a lower and steeper descent. A golf ball goes farther than this simple model predicts because its dimples and backspin change the aerodynamics, a reminder that a numerical method is only as good as the forces it is given.