Motion in Two Dimensions-Motion in a Plane - Average and Instantaneous Velocities

When an object moves on a flat surface, we describe its position by two numbers: its x-coordinate (horizontal) and y-coordinate (vertical).

Motion in two dimensions

Position Vector

• At time t₁, the object is at P with coordinates (x₁, y₁).
• We write its position vector as
  \[\vec r_1\] = x₁ \[\hat i\] + y₁ \[\hat j\].
• At time t₂, it is at Q with coordinates (x₂, y₂):
  \[\vec r_2\] = x₂ \[\hat i\] + y₂ \[\hat j\].

Displacement

• Displacement Δ\[\vec r\] is the change in position from P to Q:
  Δ\[\vec r\] = \[\vec r_2\] - \[\vec r_1\] = (x₂ - x₁)\[\hat i\] + (y₂ - y₁)\[\hat j\].

Total displacement divided by elapsed time.

The velocity at an exact moment—drawn as the tangent to the path.

\[\vec{v}_\mathrm{avg}=\frac{\Delta\vec{r}}{\Delta t}=\left(\frac{x_2-x_1}{t_2-t_1}\right)\hat{i}+\left(\frac{y_2-y_1}{t_2-t_1}\right)\hat{j}\]

Components:

• v_avg,x = \[\frac{x_2-x_1}{t_2-t_1}\]
• v_avg,y = \[\frac{y_2-y_1}{t_2-t_1}\]

Magnitude & Direction:

\[v_{\mathrm{avg}}=\sqrt{v_x^2+v_y^2},\quad\theta=\tan^{-1}\left(\frac{v_y}{v_x}\right)\]

\[\vec{v}=\lim_{\Delta t\to0}\frac{\Delta\vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}\]

Magnitude & Direction:

v = \[\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2},\quad\theta=\tan^{-1}\left(\frac{dy/dt}{dx/dt}\right)\]

Curve of motion with a tangent line at P, and component arrows showing v_x and v_y.)

Instantaneous velocity