Classical Mechanics

Vector Kinematics

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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