Introduction to Motion in a Plane

Motion in a plane refers to two-dimensional motion, where an object changes its position in both x and y directions simultaneously.

The position vector of a particle in a plane is: r⃗(t)=x(t)i^+y(t)j^\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}

The velocity vector is the derivative of the position vector: v⃗(t)=dr⃗dt=dxdti^+dydtj^=vxi^+vyj^\vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} = v_x\hat{i} + v_y\hat{j}

The acceleration vector is the derivative of velocity: a⃗(t)=dv⃗dt=axi^+ayj^\vec{a}(t) = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j}

Key Characteristics:

  • Vectors must be added using component method.
  • Uniform motion in a plane results in straight-line paths.
  • Non-uniform motion yields curved trajectories.

Example Path: Circular Motion

In circular motion, even at constant speed, velocity changes direction, hence acceleration is present.

  • Position: r⃗(t)=Rcos⁡(ωt)i^+Rsin⁡(ωt)j^\vec{r}(t) = R\cos(\omega t)\hat{i} + R\sin(\omega t)\hat{j}
  • Velocity: Always tangent to the circle.
  • Acceleration: Always directed toward the center (centripetal).

Example:

A particle moves in a plane with position vector r⃗(t)=4ti^+3t2j^\vec{r}(t) = 4t\hat{i} + 3t^2\hat{j}.

Find the velocity and acceleration at time t=2t=2 s.

Solution:

  • Velocity: v⃗(t)=dr⃗dt=4i^+6tj^\vec{v}(t) = \frac{d\vec{r}}{dt} = 4\hat{i} + 6t\hat{j} → At t=2t = 2: v⃗=4i^+12j^\vec{v} = 4\hat{i} + 12\hat{j}
  • Acceleration: a⃗(t)=dv⃗dt=6j^\vec{a}(t) = \frac{d\vec{v}}{dt} = 6\hat{j}