Projectile Motion & Vectors Test Review TAKE GOOD NOTES!! YOU MAY USE ONLY THESE NOTES ON THE TEST TOMORROW!!

Projectile motion is the motion of an object in flight including the impact of gravity The path taken in flight is known as a parabola

Marble/Ramp Vectors Marble rolls down the ramp Marble travels in the x direction Marble travels in the x direction

Vectors A vector is a drawing showing direction and magnitude 45 mph east 9.8 m/s 2 down

A Thrown ball The quarterback throws the ball at 9 m/s at a 30° angle (to horizontal) 9 m/s

A Thrown ball The ball moves in the x direction 9 m/s The ball moves in the y direction

Vectors can add together Vectors can work together to describe the final, or resultant, vector For example A boat traveling down a river gets to go faster because the river “pushes” the boat faster A boat traveling up river goes slower because it has to go against the river

River Trip Boat travels 35 mph up river River travels 10 mph downstream Result – boat travels 25 mph up river

River Trip Boat travels 35 mph down river River travels 10 mph downstream Result – boat travels 45 mph down river

Vectors can add in any direction! Boat travels 35 mph across the river River travels 10 mph downstream Resultant Boat travels 35 mph across the river River travels 10 mph downstream Resultant

You Practice – draw vectors and calculate resultant An airplane flies at 255 mph with a 45 mph tailwind (from behind) A canoeist paddles at 15 mph up river, while the river flows 3 mph the other way A swimmer swims at 6 mph across (perpendicular) a river flowing at 2 mph – use a 2 + b 2 = c 2 when vectors are at right angles to each other

You Practice – Answers An airplane flies at 255 mph with a 45 mph tailwind (from behind) 255 mph 45 mph Resultant = 300 mph

You Practice – Answers A canoeist paddles at 15 mph up river, while the river flows 3 mph the other way 15 mph3 mph Resultant = 12 mph

You Practice – Answers A swimmer swims at 6 mph across (perpendicular) a river flowing at 2 mph a 2 + b 2 = c = c 2 or 40 = c 2 6 mph 2 mph Resultant = 6.32 mph

Calculating the vectors The y vector = sin of angle X velocity sin(30)*9 m/s V y = 4.5 m/s 9 m/s 30° When the football is thrown, it goes upwards at 4.5 m/s

Calculating the vectors The x vector = cos of angle X velocity cos(30)*9 m/s V x = 7.79 m/s 9 m/s 30° When the football is thrown, it goes down field at 7.79 m/s

Vector formulas V x = cos θ * original velocity θ is the angle from horizontal V y = sin θ * original velocity

You Practice – vectors Find the x and y vectors for the football thrown as shown 12 m/s 50° V x = cos(50)*12 m/s V x = 7.71 m/s V y = sin(50)*12 m/s V y = 9.19 m/s

What if it is launched horizontally (no V y )? Here is a sample flight with no starting V y This is the marble lab we did Monday!!

Horizontal Launching A marble rolls off a table 1.5 m high with a velocity of 5 m/s How far from the table will it hit the floor? Formulas d=1/2at 2 v=d/t 5 m/s 1.5 m

Use d=1/2at 2 to find time it drops (and flies away from the table) 1.5 m = ½(9.8 m/s 2 )(t 2 ) 3m/(9.8 m/s 2 )= t 2 t 2 =.31 seconds 2 t =.56 seconds Find the distance 5 m/s=d/.56 seconds 2.8 m = d 5 m/s 1.5 m

You Practice – Horizontal Launching A marble rolls off a table 3.2 m high with a velocity of 2.5 m/s How far from the table will it hit the floor? Formulas d=1/2at 2 v=d/t 5 m/s 1.5 m

You Practice - Answers Use d=1/2at 2 to find time it drops (and flies away from the table) 3.2 m = ½(9.8 m/s 2 )(t 2 ) 6.4m/(9.8 m/s 2 )= t 2 t 2 =.65 seconds 2 t =.81 seconds Find the distance 2.5 m/s=d/.81 seconds 2.0 m = d 5 m/s 1.5 m

Using vectors A projectile has a curved path as it flies It spends half of its flight time on the way up, and half on the way down 12 m/s 50°

Using vectors Let’s find how long it takes for the ball to reach the top of its trajectory, or curved path 12 m/s 50° V x = cos(50)*12 m/s V x = 7.71 m/s V y = sin(50)*12 m/s V y = 9.19 m/s

How long does it fly? First, we know it goes up at 9.19 m/s Second, we know velocity at the top of the trajectory is 0 m/s Third, we know that the upwards velocity decreases at 9.8 m/s 2 due to gravity 12 m/s 50°

How long does it fly? Use the formula a=(V f -V o )/t -9.8 m/s 2 =(0 m/s-9.19 m/s)/t -9.8 m/s 2 =(-9.19 m/s)/t t=(-9.19 m/s)/(-9.8 m/s 2 ) t=.94 seconds to fly up t total = 2*.94 t total = 1.88 seconds 12 m/s 50°

How far it fly? First, we now know the time it flies (1.88 seconds) Second, we know the horizontal velocity (v x ) = 7.71 m/s Use the formula v=d/t 7.71 m/s = d/1.88 seconds d=7.71 m/s * 1.88 s d = m 12 m/s 50°

You Practice – Using vectors 20 m/s 25° V x = cos(25)*20 m/s V x = m/s V y = sin(25)*20 m/s V y = 8.45 m/s

You Practice – How long does it fly? Use the formula a=(V f -V o )/t -9.8 m/s 2 =(0 m/s-8.45 m/s)/t -9.8 m/s 2 =(-8.45 m/s)/t t=(-8.45 m/s)/(-9.8 m/s 2 ) t=.86 seconds to fly up t total = 2*.86 t total = 1.72 seconds 12 m/s 50°

You Practice – How far it fly? First, we now know the time it flies (1.72 seconds) Second, we know the horizontal velocity (v x ) = m/s Use the formula v=d/t m/s = d/1.72 seconds d=18.13 m/s * 1.72 s d = m 12 m/s 50°