module 5: advanced mechanics

Question 1

What is a projectile?

Answer 1

A projectile is an object that is moving freely under the influence of gravity.
* For an object to be a projectile, there must be no force other than the gravitational force acting on the object.
* The net force on the projectile is the gravitational force (its weight).

The trajectory of a projectile follows a parabolic path.

Question 2

Horizontal component of projectile motion

Answer 2

Horizontal Motion is constant velocity motion, the velocity is the same as the horizontal component of the initial velocity.

Question 3

Vertical component of projectile motion

Answer 3

Vertical motion is uniform accelerated motion, the projectile experiences a constant downward acceleration due to gravity.

Question 4

Projectile situation 1:

Launched from an elevated position and follows a parabolic path to the ground

Answer 4

• Initial velocity is only in the horizontal direction (Initial velocity = horizontal velocity).
• Velocity direction does not have initial velocity (initial vertical velocity = 0 ms-1).
• Launch angle 𝜃 = 0°

Question 5

Projectile situation 2:

Launched at an angle, rising upward to a peak before falling back down

Answer 5

• Total vertical displacement = 0
• When y is at maximum height, v = 0 ms-1
• Initial speed = final speed
• Time to rise = time to fall
• Initial angle = final angle

Question 6

Projectile situation 3:

Launched at an angle from an elevated position

Answer 6

• Total vertical displacement = difference in height between the two levels
• When y is at maximum height, v = 0 ms-1
• Time to rise ≠ time to fall
• Initial speed ≠ final speed
• Initial angle ≠ final angle

Question 6

Circular motion

Answer 6

For an object to move in a circular path, it must continually change the direction its travelling, hence circular motion requires a net force to be applied on the object.

Uniform circular motion refers to objects that move in a circle at constant speed.

Question 7

Centipetal force

Answer 7

Centripetal force is the force that causes an object to move in a circular path and is directed towards the centre of the circle.

Question 8

Tangential Velocity

Answer 8

Tangential velocity is the velocity of an object experiencing uniform circular motion.
* Tangential velocity of an object undergoing circular motion is related to the radius of the circle and the time it takes the object to complete one revolution.

Question 9

Tangential Velocity equation

Answer 9

v = 2πr/T

Where:
v is the tangential velocity (ms-1)
r is the radius (m)
T is the period of rotation i.e. the time to go around the circle once (s)

Question 10

Centripetal Accleration equation

Answer 10

ac = v^2 / r

Question 11

Centripetal Force

Answer 11

Any motion in a curved path represents accelerated motion and requires a force directed towards the centre of curvature of the path.
* This force is called centripetal force which means “centre seeking” force.

Question 12

Centripetal Force equation

Answer 12

FCentripetal = mv^2 / r

Question 13

Unbanked curve

Answer 13

On an unbanked curve, the static frictional force from the car tires provides the centripetal force.
* If the frictional force is less than the centripetal force required, then the car cannot maintain its circular path and the car will skid at a tangent to the circle.

Question 14

Banked curve

Answer 14

When the curve is banked, the centripetal force can be partially or entirely supplied by the horizontal component of the normal force.
* A reason to have banked curves would be to decrease the reliance on the force of friction.
* For every banked curve, there is one ideal speed (regardless of mass).

Question 15

Ideal speed

Answer 15

When centripetal force is entirely supplied by the horizontal component of the normal force, it is called ‘ideal speed’.
* Speed of car > ideal speed, car will slide up inclined plane.
* Speed of car < ideal speed, car will slide down inclined plane.

Question 16

Angular acceleration

Answer 16

In UCM, angular velocity (⍵) is constant.
* If ⍵ changes with time, angular speed is increasing or decreasing, therefore angular acceleration is present
* For an object to start rotating around an axis, it requires a force to give the object some angular acceleration.

Question 17

Torque

Answer 17

Torque is the rotational equivalent of linear force, it is a vector quantity.
* Occurs when a force, F, is applied at some displacement, r, from an axis of rotation
* The direction of the torque is perpendicular to the plane of rotation, and the direction is determined using the right hand
* Have to indicate whether the torque turns the object clockwise or anticlockwise

Question 18

Properties of ellipses

Answer 18

1. An ellipse is defined by two focal points, together called foci.
   * The sum of the distances to the foci from any point on the ellipse is always constant.
2. The amount of flattening of the ellipse is called the eccentricity.
   * If e = 0, it’s a circle.
   * If e = 1, it’s a parabola
   * All ellipses have eccentricity in the range: 0 ≤ e ≤ 1.
3. The longest axis of the ellipse is that major axis while the short is called the minor axis.
   * Half the major axis is termed a semi-major axis.

Question 19

Kepler’s First Law - Law of ellipses/Law of orbits

Answer 19

First law: The planet orbits the sun in elliptical orbits, with the Sun as one of the focal points.

Question 20

Kepler’s Second Law - Law of equal areas

Answer 20

Second law: the line between a planet and the sun (the radius vector) sweeps out equal areas in equal periods of time.

Question 21

Kepler’s Third Law - Law of Periods

Answer 21

Third law: The square of a planet’s period, T, is directly proportional to the cube of the semi-major axis of its orbit.

Question 22

Orbit characteristics

Answer 22

Orbits must be:
* Centred on the centre of the mass of the central object
* Elliptical or circular
Orbit characteristics are:
* Altitude (circular or near circular)
* Nearest and furthest point (e.g. perigee and apogee, if elliptical)
* Inclination (relative to the equator)

Question 23

Satellite definition

Answer 23

A satellite is an object that orbits a central body.
* A manufactured satellite is called an artificial satellite.
* Planets, asteroids, comets etc. are called natural satellites.

Question 24

Low Earth Orbits (LEO)

Answer 24

LEOs are slowed slightly by drag from the upper atmosphere, consequently called degenerate orbits as LEO satellites will eventually spiral back into the atmosphere.

Usually burns up during re-entry.

Question 25

LEO characteristics

Answer 25

Altitude: 160 and 2000 km
Orbital period: 128 minutes or less
Mean velocity to remain in orbit: 7.8 km/s
Used to measure a wide range of parameters such as ocean currents, temperature, icecap coverage, weather prediction (polar orbits) and satellite imaging.

Question 26

Middle Earth Orbits

Answer 26

Altitude: 2000 - 35786 km
Used for navigation (GPS) and communication

Question 27

Geosynchronous Earth Orbit

Answer 27

A geosynchronous orbit (GSO) is a high Earth orbit that allows satellites to match Earth’s rotation.
* Located 35,786 km above the Earth’s equator, is a valuable spot for monitoring weather, communications and surveillance.
* Has an orbital of 24 hours.

Question 28

Geostationary Earth Orbit

Answer 28

A geostationary orbit is a type of GSO that is equatorial, meaning that they remains above one point on the equator as the Earth rotates.
* Has an orbital of 24 hours.

Question 29

Escape Velocity

Answer 29

Escape velocity is the minimum velocity with which an object can escape the gravitational pull of the planet