Relative Position, Relative Velocity, and Galilean Transformations

Relative Position and Relative Velocity

Definitions and Vector Relationships

Understanding the motion of particles relative to different reference points is fundamental in classical mechanics. The concepts of relative position and relative velocity allow us to describe how one object moves with respect to another.

• Position Vector: The position of a particle P with respect to an origin O at time t is given by the vector $\mathbf{r}_{P/O}(t) = \langle x_P(t), y_P(t), z_P(t) \rangle$.

• Relative Position: The position of particle Q with respect to particle P is $\mathbf{r}_{Q/P}(t) = \mathbf{r}_{Q/O}(t) - \mathbf{r}_{P/O}(t)$.

• Component Form: $\mathbf{r}_{Q/P}(t) = \langle x_Q(t) - x_P(t),\ y_Q(t) - y_P(t),\ z_Q(t) - z_P(t) \rangle$.

Triangle Law of Vector Addition: The vectors satisfy $\mathbf{r}_{P/O}(t) + \mathbf{r}_{Q/P}(t) = \mathbf{r}_{Q/O}(t)$.

Relative Velocity

The relative velocity of Q with respect to P is the time derivative of the relative position vector:

• $\mathbf{v}_{Q/P}(t) = \frac{d}{dt} \mathbf{r}_{Q/P}(t) = \mathbf{v}_{Q/O}(t) - \mathbf{v}_{P/O}(t)$

• Where $\mathbf{v}_{P/O}(t) = \langle \frac{dx_P}{dt}, \frac{dy_P}{dt}, \frac{dz_P}{dt} \rangle$ and $\mathbf{v}_{Q/O}(t) = \langle \frac{dx_Q}{dt}, \frac{dy_Q}{dt}, \frac{dz_Q}{dt} \rangle$

Collision and Closest Approach\

Two particles collide when their relative position vector is zero: $\mathbf{r}_{Q/P}(t) = 0$.

• To find the time of collision, solve the system of equations resulting from setting each component of $\mathbf{r}_{Q/P}(t)$ to zero.

• If no common solution exists, the particles do not collide.

Example 1: Collision of Two Particles

• Given: $\mathbf{r}_{P/O}(t) = \langle 5 + 2t,\ -3 + 5t,\ 4 + 3t \rangle$ m, $\mathbf{r}_{Q/O}(t) = \langle 7 - 3t,\ 5 - 15t,\ -2 + 18t \rangle$ m

• Relative position: $\mathbf{r}_{Q/P}(t) = \langle 2 - 5t,\ 8 - 20t,\ -6 + 15t \rangle$ m

• Setting each component to zero yields $t = 2/5$ s as the collision time.

• Collision point: $\mathbf{r}_{P/O}(2/5) = \langle 29/5,\ -1,\ 26/5 \rangle$ m

• Distance from origin: $\sqrt{(29/5)^2 + (-1)^2 + (26/5)^2} = 7.85$ m

• Velocities at collision:

  • $\mathbf{v}_{P/O} = \langle 2, 5, 3 \rangle$ m/s

  • $\mathbf{v}_{Q/O} = \langle -3, -15, 18 \rangle$ m/s

  • $\mathbf{v}_{Q/P} = \langle -5, -20, 15 \rangle$ m/s

• Speeds:

  • P: $\sqrt{2^2 + 5^2 + 3^2} = 6.16$ m/s

  • Q: $\sqrt{(-3)^2 + (-15)^2 + 18^2} = 23.62$ m/s

  • Relative: $\sqrt{(-5)^2 + (-20)^2 + 15^2} = 25.50$ m/s

Example 2: Closest Approach Without Collision

• Given: $\mathbf{r}_{P/O}(t) = \langle 3 + 4t,\ 1 + t,\ 5 - 3t \rangle$ m, $\mathbf{r}_{Q/O}(t) = \langle 1 + t,\ -3 + 2t,\ 2 + 2t \rangle$ m

• Relative position: $\mathbf{r}_{Q/P}(t) = \langle -2 - 3t,\ -4 + t,\ -3 + 5t \rangle$ m

• Setting each component to zero yields no common solution; no collision occurs.

• Distance between particles: $d(t) = \sqrt{(-2 - 3t)^2 + (-4 + t)^2 + (-3 + 5t)^2}$

• To find minimum distance, minimize $d^2(t) = 35t^2 - 26t + 29$

• Set derivative to zero: $\frac{d}{dt} d^2(t) = 70t - 26 = 0 \implies t = 26/70$

• Second derivative positive, so this is a minimum.

• Minimum distance: $d_{min} = \sqrt{35(26/70)^2 - 26(26/70) + 29} = 4.92$ m

The Galilean Transformations

Reference Frames and Transformations

The Galilean transformations relate the coordinates of events as measured in two inertial reference frames moving at constant velocity relative to each other. They are foundational for Newtonian mechanics and assume absolute time.

• Let O and O' be the origins of two coordinate systems, with O' moving at constant velocity U along the x-axis relative to O.

• At time t, the position of O' relative to O is $\mathbf{r}_{O'/O}(t) = (Ut, 0, 0)$.

• Assume $t' = t$ (absolute time).

Galilean Transformation Equations

• For a particle P moving along the x-axis:

  • $x' = x - Ut$

  • $y' = y$

  • $z' = z$

  • $t' = t$

These equations allow us to transform the coordinates of any event from one inertial frame to another moving at constant velocity relative to the first.

Application: Train and Platform Example

This example illustrates how events are described in different reference frames using Galilean transformations.

• O (platform) and O' (train) are two reference frames.

• At $t = 0$, the train enters the station; clocks in both frames are synchronized ($t = t' = 0$).

• At $t_1$, the front of the train passes the observer at $x_1$ on the platform; in the train's frame, this is $x'_2$ at $t'_1$.

• At $t_2 > t_1$, the rear of the train passes the observer at $x_1$; in the train's frame, this is $x'_1$ at $t'_2$.

• The time interval $\Delta t = t_2 - t_1$ is the same in both frames due to the assumption of absolute time.

• The spatial separation in the train's frame is $\Delta x' = x'_2 - x'_1$ (length of the engine).

• To measure the length of the moving engine in the platform frame, simultaneous events at the front and rear must be chosen.

Summary Table: Galilean Transformation Variables

Quantity	O Frame (Platform)	O' Frame (Train)
Position	$x$	$x' = x - Ut$
Time	$t$	$t' = t$
Velocity	$v$	$v' = v - U$

Key Points

• Relative position and velocity are determined by vector subtraction of position and velocity vectors, respectively.

• Collisions occur when the relative position vector is zero; closest approach is found by minimizing the distance function.

• Galilean transformations relate coordinates and velocities between inertial frames, assuming absolute time and space.

• Simultaneity is absolute in Galilean relativity; events simultaneous in one frame are simultaneous in all frames.

Additional info: In modern physics, the Galilean transformations are replaced by Lorentz transformations when dealing with speeds close to the speed of light, where time and space are not absolute.