Two Dimensional Two Dimensional MotionMotion

Vector DiagramsVector Diagrams

Depict the motion of an object using Depict the motion of an object using vectors.vectors.– Remember all vectors have a Remember all vectors have a

magnitude, direction and unit.magnitude, direction and unit.– Displacement, velocity and acceleration Displacement, velocity and acceleration

are vectors.are vectors.

Magnitude Shown With Arrow Magnitude Shown With Arrow LengthLength

Vector DirectionVector Direction

Vectors Can Be Added and the Vectors Can Be Added and the Sum is Called the Sum is Called the ResultantResultant

VectorVector

Finding the Resultant VectorFinding the Resultant Vector When adding or subtracting vectors When adding or subtracting vectors

always connect the head of one vector to always connect the head of one vector to the tail of the next.the tail of the next.

The direction in which the vector is The direction in which the vector is pointing indicates the direction of the pointing indicates the direction of the velocity, acceleration or force.velocity, acceleration or force.

Vectors in opposite directions are Vectors in opposite directions are subtracted and vectors in the same subtracted and vectors in the same direction are added.direction are added.

Trigonometry must be used to find the Trigonometry must be used to find the magnitude of the resultant vector of magnitude of the resultant vector of motion in two dimensions.motion in two dimensions.

Indicating Vector DirectionIndicating Vector Direction

Magnitude of the Vector is the Magnitude of the Vector is the Length of the ArrowLength of the Arrow

Practice

• Determine the direction of this vector

Adding Vectors in 2 Dimensions

+8 meters

10 meters

Always connect vectors head to tail

or

Adding Vectors in 2 Dimensions

8 meters

10 meters

Use the Pythagorean Theorem to find the hypotenuse

82 + 102 = C2

C = 12. 8 meters

Adding vectors in 2 Dimensions

+

8 meters [N45o E] 10 meters [E]

Find the x and y components of the vector at an angle.

Add the x components and then the y components separately.

Then make a new triangle and find the new hypotenuse.This is the resultant vector

Then find the angle of declination of the resultant vector.

Adding Vectors in 2 Dimensions

8 meters [N45oE] 10 meters [E]

45o

x component y component

Vector 1 COS 45o x 8 meters [E] SIN 45o x 8 meters [N]Vector 2 10 meters [E] no y component_______________________________________________5.66 meters [E] 5.66 meters [N]10 meters [E]

15.66 meters [E] 5.66 meters [N]

Adding Vectors in 2 Dimensions

15.66 Meters [E]

5.66 meters [N]

15.662 + 5.662 = c2

The resultant vector =16.65 meters

Find the angle between the tail of the first vector drawnand the tail of the resultant vector.

TAN angle = o TAN-1 = 5.66 a 15.6616.65 meters [E20oN] or 43 o north of east

ProjectilesProjectiles

Airborne objects that move as a Airborne objects that move as a result of their own inertia and result of their own inertia and gravity.gravity.

They exhibit two dimensional motion.They exhibit two dimensional motion.– Horizontal and vertical motionHorizontal and vertical motion

Horizontal and vertical motion are Horizontal and vertical motion are independent of one another.independent of one another.

– The resultant motion is a combination of The resultant motion is a combination of horizontal and vertical motion.horizontal and vertical motion.

Components of Projectile Components of Projectile MotionMotion

Projectiles have:Projectiles have:– vertical velocities that increase and vertical velocities that increase and

decrease because they are influenced decrease because they are influenced by gravitational forces.by gravitational forces.

– horizontal velocities remain constant horizontal velocities remain constant because they are not influenced by because they are not influenced by gravitational forces.gravitational forces.

Why Does the Hunter Miss the Monkey

The Hunter Now Aims

Monkey In the Absence of Gravity

Practice• A zookeeper finds an escaped monkey hanging

from a light pole. Aiming her tranquilizer gun at the monkey, the zookeeper kneels 10 meters from the light pole which is 5 meters high. The tip of her gun is 1 meter above the ground. ground. The monkey tries to trick the zookeeper by dropping the banana, then continues to hold onto the light pole. At the moment the monkey releases the banana, the zookeeper shoots. If the tranquilizer travels at 50 m/s. Will the dart hit the monkey, the banana or neither one?

Displacement for Projectiles

Notice that the displacement in the Notice that the displacement in the horizontal direction reflects the equation:horizontal direction reflects the equation:

d = v td = v t

Notice that the displacement in vertical Notice that the displacement in vertical direction reflects the equation:direction reflects the equation:

d = vtd = vt22

This is because the projectile is accelerating to the This is because the projectile is accelerating to the ground in the vertical direction.ground in the vertical direction.

The combination of both components The combination of both components creates a parabolic path.creates a parabolic path.

Two Dimensional Horizontal Two Dimensional Horizontal Motion for ProjectilesMotion for Projectiles

Horizontal displacement “Horizontal displacement “xx” is called ” is called the range.the range.

d = d = vv11tt x = vt x = vt

– Assume Assume vv11 does NOT change. does NOT change.

– So So vv11 = = vv22 therefore the acceleration therefore the acceleration in the in the xx or horizontal direction is always or horizontal direction is always zero.zero.

Two Dimensional Vertical Two Dimensional Vertical Motion for ProjectilesMotion for Projectiles

Displacement in the “Displacement in the “yy” is called the ” is called the hieght.hieght.– Assume the acceleration is -9.8 m/sAssume the acceleration is -9.8 m/s22

– Assume Assume vv11 is zero is zero

d=1/2 atd=1/2 at2 2 y = ½ aty = ½ at22

VV2222 = 2ad V = 2ad V22

22 = 2ay = 2ay

VV22 = at V = at V22 = at = at

Strategies for Applying Strategies for Applying Kinematic Equations in 2 Kinematic Equations in 2

DimensionsDimensions Make a drawing of the situation.Make a drawing of the situation. Decide which directions to be called positive and Decide which directions to be called positive and

negative. Do this for both the x and y directions.negative. Do this for both the x and y directions. Remember that the time variable is the same for Remember that the time variable is the same for

both x and y.both x and y. Make note of all the assumed information.Make note of all the assumed information.

– Starts at rest.Starts at rest.– StopsStops

When motion is divided into into two segments, When motion is divided into into two segments, remember the final velocity for one segment remember the final velocity for one segment becomes the initial velocity for the next segment.becomes the initial velocity for the next segment.

Keep in mind there might be two possible Keep in mind there might be two possible answers form the same problem. Choose the answers form the same problem. Choose the answer that matches the physical situation at answer that matches the physical situation at hand.hand.