Physics - Projectile Motion

Galileo’s Analysis of Projectile Motion

“A projectile motion consists of two independent motions, the horizontal and vertical motion. The horizontal motion is under constant velocity and the vertical motion is under constant acceleration.”

Equations of Motion:

▪ v = u + at

▪ v²= u² + 2as

▪ s = ut + ½ at²

s = vt.

Projectile Motion Variables

▪ Initial velocity (u) ▪ Angle of projection (𝜃) ▪ Vertical and horizontal velocity (ux, uy) ▪ Maximum height (h) ▪ Change in vertical displacement (sy) ▪ Time of flight ▪ Range (sx) ▪ Final velocity (v)

Uniform Circular Motion

• Velocity changes since the direction changes.

• Instantaneous velocity is tangential to its path.

• Time for one full cycle is called a period. (T, sec)

• Number of rotations in one second is called frequency. (f, Hz) f = 1/T

Average Speed

v = (2 x (pi) x r) / T

Angular Velocity

• ω – angular velocity (0 s^-1 / 𝜋 𝑠^-1 )

• 𝜃 – angular displacement (o or 𝜋 𝑐 )

• t – time (seconds)

• ω = 𝜃 / t

angular displacement

𝒍 = 𝒓𝜽 – if in radians ▪ 𝒍 = 𝟐𝝅𝒓 ( 𝜃 / 360) – if in degrees

Centripetal Acceleration

• A change in velocity implies there is acceleration. Therefore, there must be a net force experienced by the object.

• The object deviates inwards due to an acceleration towards the centre of the circle. This is called centripetal acceleration.

• Centripetal acceleration always acts towards the centre. The object would fly off along the tangential path without acceleration.

• ac always acts towards the centre of motion.

• 𝒂𝒄 = 𝒗² / 𝒓 = 𝟒𝝅²𝒓 / T

Centripetal Force

• Responsible for circular motion.

• Fnet = ma = 𝒎𝒗² / 𝒓

• Tension, Normal, Gravity, Friction.

Banked Curves

• When a vehicle turns on a flat surface, it relies on the friction to provide the centripetal force. This may not always be present if the surface is smooth (icy surface or worn types).

• Horizontal component of the normal force provides the centripetal force on banked curves.

• Design speed is the max speed on a banked curved without the vehicle drifting higher or lower.

• tan𝜽 = mv² / r x g

• v = ( tan𝜽 x r x g)^1/2

Work

W = FS = Fscos𝜽

• W – Work (J)

• F – force (N)

• s – displacement (m)

• 𝜃 – angle between the force vector & the displacement vectors

Torque:

T = Fdxsin𝜃⊥

If the force is applied perpendicular to the pivot point, the torque is at a maximum.

Newton’s Law of Universal Gravitation

Gravitational force is an attractive force. This exists between any two objects.

F = 𝑮𝑴𝒎 / 𝒓²

• F – gravitational force (N)

• M, m – mass of objects (kg)

• r – centre to centre distance of separation (m)

• G – universal constant (6.67 x 10-11 Nm2 kg-2

Altitude vs. Orbital Radius

As Altitude increases the force decreases.

Gravitational Field Strength:

g = GM / r²

• g - field strength (ms^-2)

• G - gravitational constant

• M - mass of planet

• r - radius of planet

Satellites

LEO (250 - 1000) - easier to get up, can communicate faster due to lower altitude, need more for a wide coverage, shorter life span (air resistance - decay) - syping, hubble.

GEO (35000) - good life span, need less, has reasonable coverage. - in orbit with the earth, higher altitude can cause delays in information. - comms and weather

MEO - long life span, need minimal as wider coverage, harder and more expensive to send up.

Properties of Orbit

• Centripetal acceleration. since gravitational force provides centripetal force. Centripetal acceleration is the same as the gravitational field strength. (something to pull it in).

• Orbital Period.

Orbital Velocity

• v = 2(pi)r / T

• GMm / r² = mv² / r

  • v = (GM / r )^1/2

Where:

• v - orbital velocity

• G - universal constant

• M - mass

• r - radius

Kepler’s Third Law

Mathematical relationship between radius and period of satellites.

r³ / T² = GM / 4 (pi)²

Gravitational Potential Energy

U = GPE = − 𝑮𝑴𝒎 / 𝒓

Where,

• U – GPE (J)

• G – universal constant (6.67 x 10-11 Nm2 kg-2 )

• M – mass of the planet (kg)

• m – mass of the object (kg)

• r – centre to centre distance (m)

It is negative a work is required to pull the two masses against gravity. As it gets so far the GPE = 0 therefore we take a new reference point where infinity is 0 GPE and all GPE values before infinity are negative.

Change in GPE

Work.

W = 𝛥GPE = GPE2 – GPE1

W = 𝐺𝑀𝑚 [ (1 / 𝑟1) − (1 / 𝑟2) ]

Mechanical Energy in Gravitational Field

E = KE + U

E = - GMm / 2r

Where:

• U – GPE (J)

• KE – Kinetic Energy(J)

• G – Universal constant (6.67 x 10-11 Nm²kg-2

• M – mass of the planet (kg)

• m – mass of the object (kg)

• r – centre to centre distance (m)

Kinetic Energy of a Satellite

K = GMm / 2r

Escape Velocity

Lowest velocity an object must obtain to escape a gravitational field and go to infinity.

KEinitial – KEfinal = GPEfinal – GPEinitial

1 / 2 𝑚𝑢² = 𝐺𝑀𝑚 / 𝑟

u = √ 2𝐺𝑀 / 𝑟

• u – escape velocity (ms-1 )

• G – Universal constant (6.67 x 10-11 Nm2 kg-2

• M – mass of the planet (kg)

• r – centre to centre distance (m)

Kepler’s 1st Law

Planets move in elliptical orbits.

Kepler’s 2nd Law

A planet sweeps out the same area for every block of time. A1 = A2 as A1 moves slower and moves faster around A2 as it has a larger area.