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Classical Mechanics

# Position Vectors

A car at point $$A$$ on a straight road goes west for $$20$$ seconds, arriving at point $$B$$ which is $$200$$ m away from $$A.$$ The car then heads back to the east for $$30$$ seconds, arriving at point $$C$$ which is $$800$$ m away from $$B.$$ What is the displacement of the car from point $$A$$?

Assume that $$+$$ is east and $$-$$ is west.

One morning, you wake up and decide to take a jog through the town. You take $$54 \text{ seconds}$$ to run $$250 \text{ m}$$ straight north, then you turn right and take $$42 \text{ seconds}$$ to run $$178 \text{ m}.$$ Then you turn right again and run down the street for $$9 \text{ seconds}$$ covering $$30 \text{ m}$$ until you to stop. Assuming that the coordinates of your home are the origin $$(0, 0),$$ find the position vector of the place where you stop.

Take eastward as $$+\hat{i}$$ and northward as $$+\hat{j} .$$

A soccer player undergoes two successive displacements: $\Delta \vec{r_A} = (28 \text{ m})\hat{i} + (8\text{ m})\hat{j} \text{ followed by } \Delta \vec{r_B} = (−28\text{ m}) \hat{i}+(7\text{ m}) \hat{j}.$ What is the total displacement of the soccer player?

The motion of a creature in three dimensions can be described by the following equations for positions in $$x, y$$ and $$z$$ directions: \begin{align} x(t)&=3t^2 + 6 \\ y(t)&=- t^2 + 3t - 2 \\ z(t)&= 3t + 1. \end{align} Find the position vector of the creature at $$t = 3.$$

The Andromeda galaxy is a giant spiral cluster of stars whose mass is that of $$300$$ billion Suns. You can see it with the naked eye as a faint elongated cloud in the night sky. Inasmuch as it subtends an angle of $$4.1^{\circ}$$ and is known to be larger than our own galaxy [$$163 \times 10^3$$ light-years (units of ly) in diameter for Andromeda as compared to $$100 \times 10^3$$ light-years for our galaxy], how far away is it in light-years?

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