Motion in Two Dimensions

Position and Velocity as Vectors #

One straight line, written r = r0 + vt.

A constant velocity moves a position vector along the straight line r = r0 + vt

At time zero the object is at $r_0=\left(1,2\right)$r₀ = (1, 2), and has not yet moved.

Its velocity is v = (3, 2): every second it moves 3 across and 2 up.

Add v again for each second that passes. The positions lie on one straight line.

Written out, each coordinate is its own straight-line equation in t.

Meeting Points and Travel Times #

Crossing paths is not the same as colliding.

Two paths crossing is not a collision unless both objects are there at the same time

These two straight paths cross. That is a fact about the routes, not about the objects on them.

For a collision both coordinates must agree, and agree at the same value of t.

Solve one coordinate for t, then test the other. Here the y-coordinates disagree, so they never collide.

Paths can cross while the objects pass the crossing point hours apart.

The Closest Approach of Two Objects #

The squared gap is a quadratic in time.

The gap between two objects is a quadratic in time, so it has one lowest value

Subtract one position vector from the other to get the gap between them, as a vector in t.

Work with the square of the length: the algebra is easier, and it is smallest at the same moment.

The curve turns once. That turning point is the moment the two are nearest.

Differentiate, set the derivative to 0 to find t, then take the square root for the distance.

Velocity That Varies with Time #

Differentiate the position vector coordinatewise.

Differentiating a position vector coordinate by coordinate gives velocity, then acceleration

A thrown ball moves steadily across, while a squared term pulls the height back down.

Plot one coordinate against the other and the path appears: a parabola.

The horizontal velocity never changes, and the vertical velocity falls by 10 every second.

Circular motion: the acceleration always points back at the center of the circle.

Motion That Starts Later #

Replace t by t − k and the path is unchanged.

An object setting off k seconds later has position r(t − k), the same rule on a delayed clock

One ball is thrown at t = 0, and this rule gives where it is at every later t.

Ball 2, thrown 2 seconds later, is at time t where ball 1 was at time t − 2.

At t = 4 ball 2 is at the top, where ball 1 was two seconds earlier.

A later start subtracts: the clock inside the rule runs behind the clock outside.

The speed and the path are unchanged — only the time at which each position is reached moves.