Question 1

Given the position vectors $$r_{P/O}(t) = \langle 5 + 2t, -3 + 5t, 4 + 3t \rangle$$ and $$r_{Q/O}(t) = \langle 7 - 3t, 5 - 15t, -2 + 18t \rangle$$, what is the relative position vector $$r_{Q/P}(t)$$?

Accepted Answer

$$r_{Q/P}(t) = \langle 2 - 5t, 8 - 20t, -6 + 15t \rangle$$

Question 2

If $$v_{P/O}(t) = \langle 2, 5, 3 \rangle m/s $$and $$v_{Q/O}(t) = \langle -3, -15, 18 \rangle m/s$$, what is the relative velocity $$v_{Q/P}(t)$$?

Accepted Answer

$$v_{Q/P}(t) = \langle -5, -20, 15 \rangle m/s$$

Question 3

Two particles P and Q have position vectors $$r_{P/O}(t)$$ and $$r_{Q/O}(t). $$Under what condition will they collide?

Accepted Answer

$$r_{Q/P}(t) = 0$$

Question 4

Given $$r_{Q/P}(t) = \langle -2 - 3t, -4 + t, -3 + 5t \rangle$$, what is the minimum distance between particles P and Q?

Accepted Answer

$$\sqrt{35\left(\frac{26}{70}\$$right)^2$$ - 26\left(\frac{26}{70}\right) + 29} m$$

Question 5

If the speed of particle P at collision is $$\|v_{P/O}\| = \sqrt{2^2$ + 5^2$ + 3^2$} m/s$$, what is the numerical value rounded to two decimal places?

Accepted Answer

$$6.16 m/s$$

Question 6

Which of the following is the correct Galilean transformation for the x-coordinate?

Accepted Answer

$$x' = x - Ut$$

Question 7

According to the Galilean transformation, how does time transform between two inertial frames?

Accepted Answer

$$t' = t$$

Question 8

A train moves with constant speed $$U$$ along the x-axis. If the platform observer measures the length of the engine using simultaneous events at the front and rear, what does this measurement represent?

Accepted Answer

The length of the engine as measured in the train's frame

Question 9

If $$r_{Q/P}(t) = \langle -2 - 3t, -4 + t, -3 + 5t \rangle$$, what is the expression for the square of the distance between P and Q?

Accepted Answer

$$( -2 - 3t )^2$ + ( -4 + t )^2$ + ( -3 + 5t )^2$

Question 10

In the context of Galilean transformations, what is meant by 'absolute time'?

Accepted Answer

Time is the same for all observers regardless of their motion

Question 11

If observer O sees observer O' move along the x-axis with constant speed $$U$$, what is the position vector of O' with respect to O at time $$t$$?

Accepted Answer

$$r_{O'/O}(t) = (Ut, 0, 0)$$

Question 12

If $$v_{Q/P}(t) = v_{Q/O}(t) - v_{P/O}(t)$$, what is the relative velocity of P with respect to Q?

Accepted Answer

$$v_{P/Q}(t) = v_{P/O}(t) - v_{Q/O}(t)$$

Question 13

A platform observer and a train observer both measure the time interval $$\Delta t$$ for the engine to pass. According to Galilean relativity, what is true about their measurements?

Accepted Answer

$$\Delta t = \Delta t'$$

Question 14

If the minimum distance between two particles is found by minimizing $$\|r_{Q/P}(t)\|^2$$, which calculus method is used?

Accepted Answer

Set $$\frac{d}{dt} \|r_{Q/P}(t)\|^2 = 0$$ and use the second derivative test

Question 15

If $$r_{Q/P}(t) = 0$$ has no solution for $$t$$, what does this imply about the motion of particles P and Q?

Accepted Answer

The particles will not collide

Question 16

If the speed of particle Q at collision is $$\|v_{Q/O}\| = \sqrt{(-3)^2$ + (-15)^2$ + 18^2$} m/s$$, what is the numerical value rounded to two decimal places?

Accepted Answer

$$23.62 m/s$$

Question 17

Which of the following statements about Galilean transformations is correct?

Accepted Answer

They assume time is absolute and the same for all observers

Question 18

If $$r_{Q/P}(t) = \langle 2 - 5t, 8 - 20t, -6 + 15t \rangle$$, at what time do the particles collide?

Accepted Answer

$$t = \frac{2}{5} s$$

Question 19

In a real-world scenario, a car and a bicycle are moving along a straight road. If the car's velocity is $$v_{C/O}$$ and the bicycle's velocity is $$v_{B/O}$$, what is the velocity of the bicycle relative to the car?

Accepted Answer

$$v_{B/C} = v_{B/O} - v_{C/O}$$

Relative Position, Relative Velocity, and Galilean Transformations

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