Vectors projectile motion

VECTORSPROJECTILE MOTIONPROJECTILE MOTION

Vectors and Direction

Key Question:

How do we accurately communicate length and distance?and distance?


� A scalar is a quantity that can be completely described by one value: the magnitude. by one value: the magnitude.

� You can think of magnitudeas size or amount, including units.


� A vector is a quantity that includes both magnitude and direction.direction.

� Vectors require more than one number.

� The information “1 kilometer,

40 degrees east of north” is

an example of a vector.


� In drawing a vector as an arrow you must choose a scale.

If you walk five meters � If you walk five meters east, your displacement can be represented by a 5 cm arrow pointing to the east.


� Suppose you walk 5 meters east, turn, go 8 meters north, then turn and go 3 meters west.

� Your position is now 8 meters � Your position is now 8 meters north and 2 meters east of where you started.

� The diagonal vector that connects the starting position with the final position is called the resultant.


� The resultant is the sum of two or more vectors added together.

� You could have walked a shorter distance by going 2 m east and 8 distance by going 2 m east and 8 m north, and still ended up in the same place.

� The resultant shows the most direct line between the starting position and the final position.

Calculate a resultant vector

� An ant walks 2 meters West, 3 meters North, and 6 meters East.

� What is the displacement of the ant?

Finding Vector Components Graphically

� Draw a displacement vectoras an arrow of as an arrow of appropriate length at the specified angle.

� Mark the angle and use a ruler to draw the arrow.

Finding the Magnitude of a Vector

� When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem.

Adding Vectors

� Writing vectors in components make it easy to add them.

Subtracting Vectors

Calculate vector magnitude

� A mail-delivery robot needs to get from where it is to the mail bin on the map. map.

� Find a sequence of two displacement vectors that will allow the robot to avoid hitting the desk in the middle.

Projectile Motion

A projectile is an

object moving in

two dimensions

under the influence

of Earth's gravity;

its path is a

parabola.

by analyzing

the horizontal

and vertical

motions

Projectile Motion

separately

Projectile Motion

The speed in the x-

direction is constant; in

the y-direction the object

moves with constant

acceleration g.acceleration g.

This photograph shows two balls that

start to fall at the same time. The one

on the right has an initial speed in the

x-direction. It can be seen that vertical

positions of the two balls are identical

at identical times, while the horizontal

position of the yellow ball increases

linearly.

Projectile Motion

If an object is launched at an initial angle of θ0 with the

horizontal, the analysis is similar except that the initial

velocity has a vertical component.

Trajectory

� The path a projectile follows is called its trajectory.trajectory.

Trajectory, Range

� The trajectory of a thrown basketball follows a special type of arch-special type of arch-shaped curve called a parabola.

� The distance a projectile travels horizontally is called its range.

Projectile Motion and the Velocity Vector

� The velocity vector (v) is a way to precisely describe the speed and direction of motion.speed and direction of motion.

� There are two ways to represent velocity.

� Both tell how fast and in what direction the ball travels.

Calculate magnitude

Draw the velocity vector v = (5, 5) m/sec and calculate the magnitude of the velocity (the speed), using the Pythagorean theorem.

Components of the Velocity Vector

� Suppose a car is driving 20 meters per second.

� The direction of the vector is � The direction of the vector is 127 degrees.

� The polar representation of the velocity is v = (20 m/sec, 127°).

Calculate velocity

� A soccer ball is kicked at a speed of 10 m/s and an angle of 30 degrees.

� Find the horizontal and vertical components of the ball’s initial velocity.

Adding Velocity Components

� Sometimes the total velocity of an object is a combination of velocities.

� One example is the motion of a boat on a river.

� The boat moves with a certain velocity relative to the water.

� The water is also moving with another velocity relative to the land.

Adding Velocity Components

Calculate velocity components

� An airplane is moving at a velocity of 100 m/s in a direction 30 degrees NE relative to the air.

� The wind is blowing 40 m/s in a direction 45 degrees SE relative to the ground.

� Find the resultant velocity of the airplane relative to the ground.

Projectile Motion

� When we drop a ball from a height we know that its speed increases

Vx

Vythat its speed increases as it falls.

� The increase in speed is due to the acceleration gravity, g = 9.8 m/sec2.

Vy

x

y

Horizontal Speed

� The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force.any horizontal force.

� Since there is no force, the horizontal acceleration is zero (ax = 0).

� The ball will keep moving to the right at 5 m/sec.

Horizontal Speed

� The horizontal distance a projectile moves can be calculated according to the formula:

Vertical Speed

� The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second.

� After one second has passed, vyof the ball will be 9.8 m/sec.

� After the 2nd second has passed, vy will be 19.6 m/sec and so on.

Calculate using projectile motion

� A stunt driver steers a car off A stunt driver steers a car off a cliff at a speed of 20 meters per second.

� He lands in the lake below two seconds later.

� Find the height of the cliff and the horizontal distance the car travels.

Projectiles Launched at an Angle

� A soccer ball kicked off the ground is also a projectile, but it starts with an initial starts with an initial velocity that has both vertical and horizontal components.

*The launch angle determines how the initial velocity divides

between vertical (y) and horizontal (x) directions.

Steep Angle

� A ball launched at a steep angle will have a largehave a largevertical velocity component and a small horizontal velocity.

Shallow Angle

� A ball launched at a low angle will have low angle will have a large horizontal velocity component and a small vertical one.

Projectiles Launched at an Angle

The initial velocity components of an object launched at a velocity vo and angle θ are found by breaking the velocity into x and y components.

Range of a Projectile

� The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle.


� The range of a projectile is calculated from the horizontal velocity and the time of flight.


� A projectile travels farthest when launched at 45 degrees.


� The vertical velocity is responsible for giving the projectile its "hang" time.

Hang Time

� You can easily calculate your own hang time.

� Run toward a doorway and jump as high as you can, touching the wall or door frame.

� Have someone watch to see exactly how high you reach.� Have someone watch to see exactly how high you reach.

� Measure this distance with a meter stick.

� The vertical distance formula can be rearranged to solve for time:

Projectile Motion and the Velocity Vector

Key Question:

Can you predict the landing spot of a projectile?

Marble’s Path

Vy

Vx

t = ?Vy

x = ?

y

In order to solve “x” we must know “t”

Y = vot – ½ g t2

v t = 0 (zero)

2y = g t2

vot = 0 (zero)

Y = ½ g t2

t2 = 2y

gt = 2y

g

Vectors projectile motion

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