Motion in Two Dimensions

01Motion in Two Dimensions

Position & Displacement

Relate position vectors to changes in position and distances.

Use the same origin and axes

\mathbf r=x\mathbf i+y\mathbf j,\qquad \mathbf s=\mathbf r-\mathbf r_0

Displacement is the difference of the two position vectors.

A position vector runs from the fixed origin to the particle. Displacement runs from its initial position to its final position. The magnitude of displacement is the straight-line distance between those positions; it is not generally the distance travelled along a curved path.

02Motion in Two Dimensions

Constant Velocity

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Position changes linearly with time

\mathbf r(t)=\mathbf r_0+t\mathbf v

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03Motion in Two Dimensions

Constant Acceleration

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Apply the same interval to each component

\mathbf v=\mathbf u+t\mathbf a

\mathbf r=\mathbf r_0+t\mathbf u+\tfrac12t^2\mathbf a

\mathbf s=\tfrac12(\mathbf u+\mathbf v)t=\mathbf v t-\tfrac12\mathbf a t^2

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04Motion in Two Dimensions

Further Calculus

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Use the usual calculus rules in each component

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05Motion in Two Dimensions

Vector Differentiation

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The basis vectors remain fixed

\mathbf v=\frac{d\mathbf r}{dt},\qquad \mathbf a=\frac{d\mathbf v}{dt}

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06Motion in Two Dimensions

Vector Integration

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An integration constant is a vector

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\mathbf r(t_2)-\mathbf r(t_1)=\int_{t_1}^{t_2}\mathbf v(t)\,dt

07Motion in Two Dimensions

Speed & Direction

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Direction of motion comes from velocity

\text{speed}=|\mathbf v|=\sqrt{v_x^2+v_y^2}

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08Motion in Two Dimensions

Meetings & Separation

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A meeting requires both coordinates to match

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\text{separation}=|\mathbf r_B(t)-\mathbf r_A(t)|

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