Theory

Motion in a plane: projectile and circular motion

How position, velocity and acceleration become vectors in the plane, how projectile motion and uniform and nonuniform circular motion are described, and how velocities and trajectories change when passing from one reference frame to another in translational motion.

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Position, velocity and acceleration vectors
Motion with constant acceleration

Position, velocity and acceleration vectors

1. Position vector and trajectory

A Cartesian frame $O x y$ with unit vectors $\overset{}{^}$ and $\overset{}{^}$ is fixed in the plane of motion. The position of a point particle $P$ at time $t$ is given by the position vector

r (t) = O P = x (t) \overset{}{^} + y (t) \overset{}{^} .

The function $t \mapsto r (t)$ is the equation of motion, and the scalar functions $x (t)$ and $y (t)$ are the equations of motion of the projections of $P$ onto the two axes. The set of points occupied by $P$ as $t$ varies is the trajectory. Eliminating $t$ between $x (t)$ and $y (t)$ gives the Cartesian equation of the trajectory; for example, with $x$ and $y$ in meters and $t$ in seconds, from $x = 2 t$ and $y = t^{2}$ one obtains $t = \frac{x}{2}$ , therefore the trajectory is the parabola $y = \frac{x ^{2}}{4}$ .

2. Displacement and average velocity

The displacement between times $t$ and $t + Δ t$ is the vector $Δ r = r (t + Δ t) - r (t)$ , which goes from the initial position to the final position. The average velocity over the interval is

v_{m} = \frac{Δ r}{Δ t},

a vector with the direction and sense of the displacement, measured in $m/s$ . The displacement depends only on the initial and final positions, so its magnitude is in general smaller than the length of the stretch of trajectory traveled. After one complete revolution on a circle the displacement is zero, and therefore the average velocity is also zero.

3. Instantaneous velocity

The velocity at time $t$ is the limit of the average velocity as $Δ t \to 0$ , that is, the derivative of the position vector:

v (t) = Δ t \to 0 lim \frac{Δ r}{Δ t} = \frac{d r}{d t} = \frac{d x}{d t} \overset{}{^} + \frac{d y}{d t} \overset{}{^} .

As $Δ t$ tends to zero the chord $Δ r$ tends to the direction of the tangent line, therefore the velocity is tangent to the trajectory at the occupied point and points in the direction of motion. The magnitude $v = ∣ v ∣ = v_{x}^{2} + v_{y}^{2}$ is the speed, that is, the time derivative of the length of the arc traveled. For $r (t) = 2 t \overset{}{^} + t^{2} \overset{}{^}$ one has $v (t) = 2 \overset{}{^} + 2 t \overset{}{^}$ , and at time $t = 1 s$ the velocity has magnitude $8 m/s \approx 2.83 m/s$ .

y = \dfrac{x^{2}}{4}

On the trajectory

y = \frac{x ^{2}}{4}

the displacement

Δ r

is the chord between the positions at times

t = 1 s

and

t = 2 s

; the velocity

v

(2, 1)

is tangent to the trajectory

4. Acceleration

The average acceleration over the interval $[t, t + Δ t]$ is $a_{m} = \frac{Δ v}{Δ t}$ , and the acceleration at time $t$ is the derivative of the velocity:

a (t) = \frac{d v}{d t} = \frac{d ^{2} r}{d t ^{2}} = \frac{d v _{x}}{d t} \overset{}{^} + \frac{d v _{y}}{d t} \overset{}{^},

measured in $m/ s^{2}$ . The acceleration records every change of the velocity vector, both in magnitude and in direction; therefore a point that travels along a curve with constant speed has nonzero acceleration. For $r (t) = 2 t \overset{}{^} + t^{2} \overset{}{^}$ one obtains $a = 2 \overset{}{^} m/ s^{2}$ , constant, while the velocity changes both in magnitude and in direction.

Motions of the projections

Since derivatives are computed component by component, the relations $v_{x} = \frac{d x}{d t}$ and $a_{x} = \frac{d v _{x}}{d t}$ involve only quantities along the $x$ axis, and the same holds along the $y$ axis. Therefore motion in a plane is equivalent to two simultaneous straight-line motions, that of the projection on $x$ and that of the projection on $y$ , which have only time in common. On each axis the laws of one-dimensional kinematics hold, for which see the topic on kinematics in one dimension. Time is the variable that links the two motions, so it is generally obtained from an equation on one axis and substituted into the equation of the other.

Motion with constant acceleration

Equations of motion with constant acceleration

Let $a$ be constant and let $r_{0}$ and $v_{0}$ be the position and velocity at time $t = 0$ . Then for every $t$

v (t) = v_{0} + a t, r (t) = r_{0} + v_{0} t + \frac{1}{2} a t^{2} .

In components there are two uniformly accelerated motions, $x (t) = x_{0} + v_{0 x} t + \frac{1}{2} a_{x} t^{2}$ and $y (t) = y_{0} + v_{0 y} t + \frac{1}{2} a_{y} t^{2}$ , and on each axis the time-independent relation $v_{x}^{2} = v_{0 x}^{2} + 2 a_{x} (x - x_{0})$ holds, and similarly for $y$ . The formulas describe every motion with constant acceleration; the most important case is the motion of bodies near the Earth’s surface, with $a = - g \overset{}{^}$ .

Proof

By definition of acceleration $\frac{d v _{x}}{d t} = a_{x}$ , with $a_{x}$ constant. By the fundamental theorem of calculus

v_{x} (t) - v_{0 x} = \int_{0}^{t} a_{x} d t^{'} = a_{x} t,

therefore $v_{x} (t) = v_{0 x} + a_{x} t$ . Since $\frac{d x}{d t} = v_{x}$ , integrating again one obtains

x (t) - x_{0} = \int_{0}^{t} (v_{0 x} + a_{x} t^{'}) d t^{'} = v_{0 x} t + \frac{1}{2} a_{x} t^{2} .

The same computation holds for the $y$ component, and recombining the components with the unit vectors gives the vector formulas. For the time-independent relation, if $a_{x} \neq = 0$ one obtains $t = \frac{v _{x} - v _{0 x}}{a _{x}}$ from the velocity law and substitutes it into the position law; carrying out the computation one obtains $x - x_{0} = \frac{v _{x}^{2} - v _{0 x}^{2}}{2 a _{x}}$ , which is the claim. If $a_{x} = 0$ the relation reduces to $v_{x} = v_{0 x}$ , which is true.

Shape of the trajectory

With constant nonzero acceleration the trajectory is a line segment or an arc of a parabola. If $v_{0}$ is zero or parallel to $a$ , the motion takes place along the line through the initial point with the direction of $a$ , and it is a uniformly accelerated straight-line motion. If $v_{0}$ and $a$ are not parallel, the $y$ axis is chosen parallel to $a$ ; then $a_{x} = 0$ and $v_{0 x} \neq = 0$ , therefore $x$ is a first-degree function of $t$ and $y$ a second-degree function. Solving for $t$ from $x$ and substituting, $y$ turns out to be a second-degree polynomial in $x$ , so the trajectory is a parabola with axis parallel to $a$ . Projectile motion is the main example of this case.

The theory continues with 6 more sections (projectile motion, maximum height time of flight and range, launches from a height…), plus all worked examples, the solutions and the PDF.

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