Two Dimensional Motion

Vector Diagrams Depict the motion of an object using vectors. Depict the motion of an object using vectors. –Remember all vectors have a magnitude, direction and unit. –Displacement, velocity and acceleration are vectors.

Magnitude Shown With Arrow Length

Vector Direction

Vectors Can Be Added and the Sum is Called the Resultant Vector

Finding the Resultant Vector When adding or subtracting vectors always connect the head of one vector to the tail of the next. When adding or subtracting vectors always connect the head of one vector to the tail of the next. The direction in which the vector is pointing indicates the direction of the velocity, acceleration or force. The direction in which the vector is pointing indicates the direction of the velocity, acceleration or force. Vectors in opposite directions are subtracted and vectors in the same direction are added. Vectors in opposite directions are subtracted and vectors in the same direction are added. Trigonometry must be used to find the magnitude of the resultant vector of motion in two dimensions. Trigonometry must be used to find the magnitude of the resultant vector of motion in two dimensions.

Indicating Vector Direction

Magnitude of the Vector is the Length 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 = C 2 C = meters

Adding vectors in 2 Dimensions + 8 meters [N45 o 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 [N45 o E]10 meters [E] 45 o x component y component Vector 1 COS 45 o x 8 meters [E] SIN 45 o x 8 meters [N] Vector 2 10 meters [E]no y component _______________________________________________ 5.66 meters [E] 5.66 meters [N] 10 meters [E] meters [E] 5.66 meters [N]

Adding Vectors in 2 Dimensions Meters [E] 5.66 meters [N] = c 2 The resultant vector =16.65 meters Find the angle between the tail of the first vector drawn and the tail of the resultant vector. TAN angle = o TAN -1 = 5.66 a meters [E20 o N] or 43 o north of east

Projectiles Airborne objects that move as a result of their own inertia and gravity. Airborne objects that move as a result of their own inertia and gravity. They exhibit two dimensional motion. They exhibit two dimensional motion. –Horizontal and vertical motion  Horizontal and vertical motion are independent of one another. –The resultant motion is a combination of horizontal and vertical motion.

Components of Projectile Motion Projectiles have: Projectiles have: – vertical velocities that increase and decrease because they are influenced by gravitational forces. –horizontal velocities remain constant because they are not influenced by 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 horizontal direction reflects the equation: Notice that the displacement in the horizontal direction reflects the equation:  d = v t Notice that the displacement in vertical direction reflects the equation: Notice that the displacement in vertical direction reflects the equation:  d = vt 2  This is because the projectile is accelerating to the ground in the vertical direction. The combination of both components creates a parabolic path. The combination of both components creates a parabolic path.

Two Dimensional Horizontal Motion for Projectiles Horizontal displacement “x” is called the range. Horizontal displacement “x” is called the range. d = v 1 t x = vt d = v 1 t x = vt –Assume v 1 does NOT change. –So v 1 = v 2 therefore the acceleration in the x or horizontal direction is always zero.

Two Dimensional Vertical Motion for Projectiles Displacement in the “y” is called the hieght. Displacement in the “y” is called the hieght. –Assume the acceleration is -9.8 m/s 2 –Assume v 1 is zero d=1/2 at 2 y = ½ at 2 V 2 2 = 2ad V 2 2 = 2ay V 2 = at V 2 = at

Strategies for Applying Kinematic Equations in 2 Dimensions Make a drawing of the situation. Make a drawing of the situation. Decide which directions to be called positive and negative. Do this for both the x and y directions. Decide which directions to be called positive and negative. Do this for both the x and y directions. Remember that the time variable is the same for both x and y. Remember that the time variable is the same for both x and y. Make note of all the assumed information. Make note of all the assumed information. –Starts at rest. –Stops When motion is divided into into two segments, remember the final velocity for one segment becomes the initial velocity for the next segment. When motion is divided into into two segments, remember the final velocity for one segment becomes the initial velocity for the next segment. Keep in mind there might be two possible answers form the same problem. Choose the answer that matches the physical situation at hand. Keep in mind there might be two possible answers form the same problem. Choose the answer that matches the physical situation at hand.