Lecture 4, Projectile Motion - Describing Movements Through the Air

Question 1

Projectile

Answer 1

a body projected into the air or dropped from a height (the object is in the air and launched/pushed/shoved/ - initial action that launches it in the air and after that there is no other force acting on it except for gravity and air resistance)
- a basketball, a discus, a high jumper, and sky diver are examples
- any object that is moving through the air unassisted

projectiles are only under the influence of air resistance and gravity (always present when an object leaves the ground)
- if air resistance is too small to measure (i.e. 0 m/s2), the only force acting on a projectile is the force of the earth’s gravity
- the acceleration due to gravity is 9.81 m/s2 downward (or -9.81 m/s2)
- the force of gravity is constant so the acceleration of the system will not change (uniform acceleration)
- air resistance only affects left/right and mainly gravity affects up/down (break it down into components so we look at them separately) - for our purpose we are going to assume there is no wind (horizontal acceleration is 0 m/s2)

Question 2

Kinematics of a Projectile

Answer 2

vertical and horizontal components are analyzed separately (since they affect different things)
1. this is because the two components are independent of each other
- gravity only influences vertical motion (up/down)
- air resistance only influences horizontal motion (forward/back)

2. once a body is projected into the air, velocity is constantly changing
   - however, horizontal and vertical velocity change predictably
   - velocity can change (things can get faster/slower) but acceleration does not (stays the same all throughout)

Question 3

Vertical Motion of a Projectile

Answer 3

• the ball has to have a velocity while it leaves/at the start the hand or else it would not go anywhere - the moment it leaves the hand gravity acts on it and pushes it down - for a split second the ball freezes at the top and highest point (apex) - as it falls from the apex gravity is pulling it down
• positive displacement means up and negative displacement means down (there is no difference between the start and end) - if you were to not catch the ball it would go below your starting point and go in the negatives
• velocity is referred to as initial velocity as it leaves the hand - first value is going to be positive as it going up but it is going to decrease and get smaller and smaller as it is going down (when it is on the line then it is at apex and freezes for a second)
• acceleration does not change so one straight line because the influence of gravity does not change from start to beginning (line is negative as gravity is negative because it pulls us down) - constant

Question 4

Vertical Motion of a Projectile (terminal velocity and drag force)

Answer 4

• projectiles continue to accelerate while in free fall
• as the speed of the system increases, an upward air resistance is created
  ◦ this drag force (a type of friction) also acts on the system
• eventually, the drag force will equal the gravitational pull on the system
• at this point the object ceases to accelerate and continues falling at a constant speed called terminal velocity (or settling velocity)
• terminal velocity is affected by the projected area
  ◦ the terminal velocity of a skydiver in free fall is: 195 km/hr with the limbs extended and 320 km/hr with the limbs held close to the body
• terminal does not mean end of movement or fastest you can go - terminal velocity means that upward going drag force matches downward going force so it falls at a constant speed no longer accelerating (question on test) - fastest in a specific position without acceleration so not necessarily the fastest you can go
• as you move through the air you are moving through air molecules and as you speed up and start to go really fast now you can feel the air where it impacts your movement
• as you start to go faster and faster the drag force equals gravity

Question 5

Horizontal Motion of a Projectile

Answer 5

• gravity does not affect the horizontal motion
• displacement is a straight line going up as it is going further and further away and it is positive because it is a rightward movement (would be opposite if we moved left)
• if we have no acceleration then velocity is a straight line and it is positive because we are going to right (there is no wind so no horizontal acceleration meaning velocity is a flat line) - we need to have a number for velocity or else there is no movement
• acceleration is just a flat line over the 0 as there is no wind

Question 6

Describing Projectile Motion

Answer 6

• trajectory = the path travelled
• apex = the highest point in the path (v = 0 m/s)
• initial velocity at the start to take off where it cannot be 0 m/s
• range is the distance between the starting and end point (horizontal displacement) - the maximum horizontal displacement travelled
• acceleration is constant (av = -9.81 m/s2 and ah = 0 m/s2)
• horizontal or total time is twice the vertical time (if you know vertical time (takeoff to peak or peak to ground) then by default you know horizontal time (takeoff to landing)

Question 7

Factors Affecting Projectile Motion

Answer 7

relative projection height, projection angle and projection speed

Question 8

If the goal is to achieve maximum height

Answer 8

• the angle of projection is 90˚
• projectile travels straight up / down (vertical trajectory)
• if all you care about is moving up to achieve maximum height (there is only up/down component)

Question 9

If the goal is to translate AND maximize the flight time in the air

Answer 9

• initial vertical velocity is greater than the horizontal velocity
• the angle of projection is above 45˚
• projectile climbs to a point then begins to fall (oblique trajectory)
• you are trying to stay in air for as long as you can with both up/down and left/right components

Question 10

If the goal is to minimize the flight time in the air

Answer 10

• the angle of projection is relatively small (i.e., below 45˚)
• if the projection angle is 0˚, the system simply drops (horizontal trajectory)
• very shallow angle is you do not want to be in air for that long
• horizontal trajectory does not only mean left/right because you up/down for trajectory (an object simply drops so there is only a down portion and left/right no up)

Question 11

Factors Affecting Projectile Motion (angles)

Answer 11

range-release angle theory (take a look at where you are starting and landing to create the optimal angle)
- optimal angle is 45 degrees = if take off and landing heights are equal
- optimal angle is > 45 degrees = if landing is above the release height
- optimal angle is < 45 degrees = if landing is below the release height

Question 12

The Laws of Constant Acceleration

Answer 12

• remember…the force of gravity is constant so the acceleration of the system will not change (uniform acceleration)
• remember…horizontal and vertical components of projectile motion are independent of each other
  laws of constant acceleration
  v2 = v1 + at
  v22 = v12 + 2ad
  d = v1t + (½)at2
• the horizontal OR vertical kinematics of projectile could be described using these formulas, depending upon the item of interest
  when you analyze projectiles analyze the horizontal and vertical separately