#2 Motion in Space | Rocket Propulsion

Quantities Used to Describe Motion

The lecture reviews the basic quantities used to analyze motion:

• Length is measured in metres. The lecturer says the metre’s standard has been based on the speed of light in vacuum since 1982.
• Mass is measured in kilograms. At the time of the lecture, its standard was still tied to a physical object, although research was underway to define it using a physical constant.
• Time is measured in seconds. The lecturer also introduces the thermodynamic arrow of time: time progresses in the direction in which entropy increases.

Velocity is the change in position per unit time, and acceleration is the change in velocity per unit time.

Momentum is defined as

[ \mathbf{p}=m\mathbf{v}, ]

and is a vector measured in kg·m/s. The lecturer presents momentum as more useful than velocity alone for analyzing impacts: an iron ball and a feather moving at the same speed have different effects when they hit something because their momenta differ.

Impulse and Force

An impulse is a change in momentum:

[ \mathbf{I}=\Delta\mathbf{p} =\mathbf{p}(t+\Delta t)-\mathbf{p}(t). ]

Impulse has the same units as momentum. During a change in momentum, the rate of change can vary over time. The lecturer describes force as this rate of change and emphasizes that, over a finite interval, it is useful to consider the average force:

[ \mathbf{F}_{\text{avg}} =\frac{1}{\Delta t}\int_t^{t+\Delta t}\frac{d\mathbf{p}}{dt}\,dt =\frac{\Delta\mathbf{p}}{\Delta t}. ]

Therefore,

[ \mathbf{I}=\mathbf{F}_{\text{avg}}\Delta t=\Delta\mathbf{p}. ]

Force describes how momentum changes; it is not interchangeable with momentum or velocity.

Universal Law of Gravitation

Building on Newton’s comparison between falling objects and planets, the lecturer introduces the gravitational attraction between two masses:

[ \mathbf{F}=-G\frac{m_1m_2}{r^3}\mathbf{r}, ]

where (r) is the separation between the masses and the minus sign indicates attraction.

The gravitational constant is given as approximately

[ G=6.671\times10^{-11}\ \text{N·m}^2/\text{kg}^2. ]

Since force is measured in newtons, with (1\text{ N}=1\text{ kg·m/s}^2), the units of (G) follow from the gravitational law.

Gravity and Trajectory Changes

The lecture connects gravitation to spaceflight and trajectory design. A sufficiently massive object placed near an asteroid could attract it and gradually alter its path. The lecturer calls this concept a gravity tractor. It illustrates that gravity can change a trajectory without direct rocket thrust.

In an example involving the Rosetta spacecraft and asteroid Steins, the lecturer estimates the attraction at the spacecraft’s closest approach. Using a spacecraft mass of about 500 kg and a centre-to-centre distance of about 1,400 km, the force is estimated to be on the order of a micronewton. The example shows that even weak forces can matter in space.

Gravitational Field and Acceleration Near Earth

For an object of mass (m) at height (h) above Earth, the lecture applies the universal law of gravitation using Earth’s mass and radius. When (h) is small compared with Earth’s radius, the force is approximated as

[ F\approx-\frac{GM_E}{R_E^2}m=-mg, ]

where (g\approx9.81\ \text{m/s}^2) near Earth’s surface.

The lecturer explains that (g) can be viewed as the gravitational field strength produced by Earth. Its units are those of acceleration, which is why it is commonly called the acceleration due to gravity.

Limits of the Gravitational Model

The lecture describes Newton’s law as based on observed phenomena and valid under particular conditions, such as ordinary cases involving separated masses and relatively low speeds. It is not sufficient for situations involving speeds close to the speed of light.

The lecturer uses a stretched sheet with a heavy ball placed in its centre as an analogy: the sheet’s deformation guides a smaller ball toward the heavy one. The lecturer cautions that this is only an analogy, not a full explanation of why gravity exists.

Frames of Reference

Motion and velocity must be described relative to a frame of reference. The lecturer points out that someone on Earth is moving because Earth rotates. The stated surface speed is about 0.46 km/s, or roughly 1,600 km/h, illustrating that an observer on Earth is not absolutely stationary.

An inertial frame of reference is described as one that is stationary or moving at constant velocity. In such a frame, changes in velocity or momentum can be assessed without additional effects associated with an accelerating or rotating frame.

Because Earth rotates, measurements made in an Earth-fixed frame can require corrections. The lecture says these issues will be developed further in a later class.

Flame Speed and Burning Velocity

The lecturer uses combustion terminology to show why the observer’s frame matters:

• Burning velocity is the flame’s motion relative to the unburned gases.
• Flame speed is the flame’s motion as observed from a stationary external frame.

If the gases are moving, flame speed and burning velocity are not generally the same. The flame’s observed speed combines the gas motion with the flame’s propagation relative to the gases; the precise relationship depends on the direction of motion.

Main Takeaway

The lecture links motion, momentum, impulse, force, gravitation, and reference frames. It emphasizes that momentum change is central to describing how motion is altered, that gravity can affect spacecraft and asteroid trajectories, and that any statement about velocity or motion depends on the frame from which it is measured.

Speakers and Sources Mentioned

• Speaker: One lecturer, whose name is not given in the subtitles.
• People and scientific figures referenced: Lucian, Johannes Kepler, Isaac Newton, Albert Einstein, Stephen Hawking, and Friedman.
• Other examples or entities referenced: The Rosetta spacecraft, asteroid Steins, Earth, the Sun, and the proposed gravity-tractor concept.