07 Key Concepts for Astronomy - Making Sense of the Universe

Describing Motion

• Speed: Rate of change of position per unit time. Scalar quantity. $speed = \frac{distance travelled}{time}$

• Velocity: Magnitude (speed) and direction of motion. Vector quantity.

• Acceleration: Rate/direction of change of velocity. Vector quantity.

• All falling objects accelerate at the same rate if there is no friction due to air resistance.

• On Earth, the gravitational acceleration is $g ≈ 9.8 m/s^2 ≈ 10 m/s^2$.

• Galileo showed that gravitational acceleration ($g$) is the same for all falling objects, regardless of their mass.

Laws of Motion

• Aristotle's ideas about motion were incorrect.

• Galilean Principle of Relativity: The fundamental laws of physics are the same, irrespective of the velocities of the observers, provided that there is no acceleration.

• Galileo’s Principle of Inertia: A body moving on a smooth (frictionless) level surface will continue in the same direction at constant speed unless disturbed.

• Newton's First Law: An object either remains at rest or in motion at constant velocity unless a net force acts on it.

• Newton's Second Law: $F = ma$ (Force equals mass times acceleration).

• Newton's Third Law: For every force, there is an equal and opposite reaction force.

Newton’s Law of Gravitation

• Every mass attracts every other mass.

• Attraction is directly proportional to the product of their masses.

• Attraction is inversely proportional to the square of the distance between their centers.

• $F<em>g = G \frac{M</em>1M_2}{d^2}$, where $G = 6.67 x 10^{-11} m^3 / kg s^2$

• Gravitational acceleration depends on the distance from Earth but is independent of a falling object’s mass: $a = g = \frac{F}{m} = G \frac{Mm}{d^2m} = G \frac{M}{d^2}$

Uniform Circular Motion

• An object in uniform circular motion constantly accelerates in the inward “radial” direction.

• Tangential velocity has constant magnitude but changing direction.

• Centripetal acceleration: $a = \frac{v^2}{r}$

• Centripetal Force: $F = m \frac{v^2}{r}$

Newton’s and Kepler’s Laws

• Newton showed that Kepler’s 1st and 2nd laws could be generalized to describe the motion of all objects, not just orbiting planets.

• Objects may move in bound (circles or ellipses) or unbound (parabola or hyperbola) paths.

Kepler’s Third Law (Newton’s Version)

• $p^2 = \frac{4π^2}{G(M<em>1 + M</em>2)} r^3$

• Where:

  • $p$ = orbital period (sec)

  • $r$ = average orbital distance between centers (m)

  • $(M<em>1 + M</em>2)$ = sum of object masses (kg)

  • $G$ = 6.67✕10-11 Nm2/kg2

Mass vs. Weight

• Mass: The amount of matter in an object.

• Weight: The force of gravity on an object.

• Objects are weightless in free-fall.