B. Kinematics in 2-Dimensions I. Mechanics B. Kinematics in 2-Dimensions

Vectors & Scalars Vectors are quantities with Magnitude AND Direction Ex: Displacement Velocity Acceleration Force Scalars are quantities with only magnitude Ex: Distance Speed Time Mass

θ Vectors Vector notation: The length of the arrow represents magnitude The direction of the vector, found from the angle, represents direction θ

Vectors

Vectors A2 + B2 = C2

Vector Practice A surveyor stands on a riverbank directly across the river from a tree on the opposite bank. She then walks 100 m downstream, and determines that the angle from her new position to the tree on the opposite bank is 50°. How wide is the river, and how far is she from the tree in her new location?

One More Practice Problem You are standing at the very top of a tower and notice that in order to see a manhole cover on the ground 50 meters from the base of the tower, you must look down at an angle 75° below the horizontal. If you are 1.80 m tall, how high is the tower?

B. Kinematics in 2-Dimensions: Projectiles

1. Projectiles A projectile is any object that falls through the air. Its path is called a trajectory. Forces acting on projectiles are: Gravity Air Resistance (which we’ll ignore because it’s small)

2. Projectile Motion The projectile’s velocity must be broken into x- and y- components Vectors that are perpendicular to each other act independently!

2. Projectile Motion Vectors that are perpendicular to each other act independently Horizontal components and vertical components don’t affect each other! We still have to make a few assumptions: g = 9.81 m/s2 (down) air resistance is negligable no horizontal a rotation of Earth can be ignored

Projectile Motion vy = v sin  vx = v cos  The velocity of a projectile has two components, vx and vy.

Practice Find the x- and y-components of the following vectors R = 175 meters @ 95o v = 25 m/s @ -78o a = 2.23 m/s2 @ 150o

x and y components of the velocity of a projectile Remember: horizontal velocity is always CONSTANT! There is NO acceleration in the x-direction! Acceleration in the y-direction is due to gravity

Velocity vectors with x- and y-components 3. General Launch Angle A projectile launched upward at an angle would have a parabolic path It moves horizontally at constant speed It accelerates downward at a rate of g = 9.81 m/s2 Velocity vectors with x- and y-components

Practice A flagpole ornament falls off the top of a 25.0 m flagpole. How long would it take to hit the ground? A stone is thrown horizontally from the top of a cliff that is 44.0 m high. It has a horizontal velocity of 15.0 m/s. We want to find how long it takes the stone to fall to the deck and how far it will travel from the base of the cliff. Tip: break each problem into two separate problems

Flagpole Problem: We assume that the ornament has no horizontal velocity. Plug in givens:

Stone problem: This is like the flagpole problem, except that the stone has an initial horizontal velocity. But we know that the time it takes to hit the ground is the same as if it were falling straight down. (This is the key concept!!!)

Stone Problem cont’d: Now that we know how long it takes to fall, we can figure out the horizontal distance it travels before it hits the ground. It has a constant horizontal speed, and can travel sideways at this speed as long as it is in the air falling, so to find x we use its average velocity and the time:

More Practice: A B-17 (a World War II era multiengine bomber) is flying at 375 km/h. The bombs it drops travel a horizontal distance of 5 250 m. What was the altitude of the plane at the time they had the old "bombs away"?

Bomber Problem: We have to find the time for the bomb to travel a horizontal distance of 5 250 m: First we convert the bomber’s speed to meters per second: Then we can find the time it takes the bomb to travel a horizontal distance of 5 250 m:

Bomber Problem cont’d: Now we can find the vertical distance (altitude):

4. Upwardly Moving Projectiles A projectile (with no air resistance) will always have a horizontal velocity that remains constant. If it is shot upward, gravity will slow down its vertical velocity

Upwardly Moving Projectiles At the top of the path, vy = 0 At the bottom of the arc, final vertical velocity = initial vertical velocity in opposite directions. The projectile travels a horizontal distance of x, also called the range.

Practice A ball is given an initial velocity of 22. 7 m/s at an angle of 66.0 to the horizontal. Find how high the ball will go. To solve this problem, we have to find the vertical velocity of the ball. Once we know it, we can find how high it goes.

The ball starts out with vy and rises till its vertical velocity is zero. We can use these as the initial and final velocity of the ball. Note that we have to pay attention here to the sign of the motion. We have both down and up motion and have to be clear about which direction is possible. Of course if the motion is in only one direction, we don’t have to worry about it.

First find the vertical velocity: A naval gun fires a projectile. The gun’s muzzle velocity (so speed of the bullet) is 345 m/s at an elevation of 32.0 . What is the range of the shot? First find the vertical velocity: Now we can find the time:

Now we can find the range since we know the time Now we can find the range since we know the time. First we find the horizontal velocity:

5. Projectile Motion Characteristics If you’re better at memorization than you are at math, well you probably should’ve chosen a different science, but here are a couple helpful formulas: Range: ymax: