Projectile Motion & Vectors

Projectile motion is the motion of an object in flight including the impact of gravity The curved flight of a football or

baseball seen during a game is an excellent example of projectile motion

Understanding Projectile Motion To understand projectile motion, it is

easiest to break the motion down into motion in two directions- x and y

We did this last unit with the marble/ramp lab The marble rolling down the ramp at an

angle was moving in the x direction (horizontally) and y direction (vertically)

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 Magnitude is the measure of how

large something is Ex. Magnitude 4.5 earthquake, 55 mph

A vector sounds a lot like a velocity, and vectors are frequently drawn to represent velocities

Drawing vectors A vector is drawn with an arrow

pointing the direction of the vector The arrow should be drawn to a

scale length Perhaps 1 cm line represents 10m/s

Vector examples

45 mph east

9.8 m/s2

down

What good are they? Vectors give a visual

representation of the directions of motion

Vectors can work together to show us what the net result will look like

Vectors can be added to calculate the net result (resultant vector)

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

Calculating the vectors

The y vector = sin of angle X velocity sin(30)*9 m/s Vy = 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 Vx = 7.79 m/s

9 m/s

30°

When the football is thrown, it goes down field at 7.79 m/s

Practicing vectors

Find the x and y vectors for the football thrown as shown

12 m/s

50°

Vx = cos(50)*12 m/sVx = 7.71 m/sVy = sin(50)*12 m/sVy = 9.19 m/s

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°

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/s2 due to gravity

12 m/s

50°

How long does it fly? Use the formula

a=(Vf-Vo)/t -9.8 m/s2=(0 m/s-9.19 m/s)/t -9.8 m/s2=(-9.19 m/s)/t t=(-9.19 m/s)/(-9.8 m/s2) t=.94 seconds to fly up ttotal = 2*.94 ttotal = 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 (vx) = 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 = 14.49 m

12 m/s

50°

Try another (simpler) 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=1/2at2

v=d/t

5 m/s

1.5 m

Try another (simpler) Use d=1/2at2

to find time it drops (and flies away from the table) 1.5 m = ½(9.8 m/s2)(t2) 3m/(9.8 m/s2)= t2

t2 = .31 seconds2

t =.56 seconds Find the distance

5 m/s=d/.56 seconds 2.8 m = d

5 m/s

1.5 m