Vectors and Direction

Vectors and Direction

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

• 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 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 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?

7.1 Finding Vector ComponentsGraphically

• Draw a displacement vector as an arrow of appropriate length at the specified angle.

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

7.1 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.

7.1 Adding Vectors

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

7.1 Subtracting Vectors

7.1 Calculate vector magnitude

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

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

7.2 Projectile Motion and the Velocity Vector

• Any object that is moving through the air affected only by gravity is called a projectile.

• The path a projectile follows is called its trajectory.


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

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


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

• There are two ways to represent velocity.

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

7.2 Calculate magnitude

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

7.2 Components of the Velocity Vector

• Suppose a car is driving 20 meters per second.

• The direction of the vector is 127 degrees.

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

7.2 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.

7.2 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.

7.2 Adding Velocity Components

7.2 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.

7.2 Projectile Motion

• When we drop a ball from a height we know that its speed increases as it falls.

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

Vx

Vy

x

y

7.2 Horizontal Speed

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

7.2 Horizontal Speed

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

7.2 Vertical Speed

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

• After one second has passed, vy of the ball will be 9.8 m/sec.

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

7.2 Calculate using projectile motion

• 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.

7.2 Projectiles Launched at an Angle

• A soccer ball kicked off the ground is also a projectile, but it 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.

7.2 Steep Angle

• A ball launched at a steep angle will have a large vertical velocity component and a small horizontal velocity.

7.2 Shallow Angle

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

7.2 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.

7.2 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.

7.2 "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.• Measure this distance with a meter stick.• The vertical distance formula can be rearranged to solve for time:


Key Question:Can you predict the landing spot of a projectile?

*Students read Section 7.2 BEFORE Investigation 7.2

Marble’s Path

Vy

x = ?

y

Vx

t = ?

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

Y = vot – ½ g t2

2y = g t2

vot = 0 (zero) Y = ½ g t2

t2 = 2y

g

t = 2y

g

7.3 Forces in Two Dimensions

• Force is also represented in x-y components.

7.3 Force Vectors

• If an object is in equilibrium, all of the forces acting on it are balanced and the net force is zero.

• If the forces act in two dimensions, then all of the forces in the x-direction and y-direction balance separately.

7.3 Equilibrium and Forces

• It is much more difficult for a gymnast to hold his arms out at a 45-degree angle.

• To see why, consider that each arm must still support 350 newtons vertically to balance the force of gravity.


• Use the y-component to find the total force in the gymnast’s left arm.


• The force in the right arm must also be 495 newtons because it also has a vertical component of 350 N.


• When the gymnast’s arms are at an angle, only part of the force from each arm is vertical.

• The total force must be larger because the vertical component of force in each arm must still equal half his weight.

7.3 Forces and Inclined Planes

• An inclined plane is a straight surface, usually with a slope.

• Consider a block sliding down a ramp.

• There are three forces that act on the block:– gravity (weight).– friction– the reaction force acting

on the block.


• When discussing forces, the word “normal” means “perpendicular to.”

• The normal force acting on the block is the reaction force from the weight of the block pressing against the ramp.


• The normal force on the block is equal and opposite to the component of the block’s weight perpendicular to the ramp (Fy).


• The force parallel to the surface (Fx) is given by

Fx = mg sinθ.

7.3 Acceleration on a Ramp

• Newton’s second law can be used to calculate the acceleration once you know the components of all the forces on an incline.

• According to the second law:

a = F m

Force (kg . m/sec2)

Mass (kg)

Acceleration (m/sec2)


• Since the block can only accelerate along the ramp, the force that matters is the net force in the x direction, parallel to the ramp.

• If we ignore friction, and substitute Newtons' 2nd Law, the net force is:

Fx =

a =

m sin θ

g

Fm


• To account for friction, the horizontal component of acceleration is reduced by combining equations:

Fx = mg sin θ - m mg cos θ


• For a smooth surface, the coefficient of friction (μ) is usually in the range 0.1 - 0.3.

• The resulting equation for acceleration is:

7.3 Calculate acceleration on a ramp

• A skier with a mass of 50 kg is on a hill making an angle of 20 degrees.

• The friction force is 30 N.• What is the skier’s acceleration?

7.3 Vectors and Direction

Key Question:How do forces balance in

two dimensions?

*Students read Section 7.3 BEFORE Investigation 7.3

Application: Robot Navigation

Vectors and Direction

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