Kinematics in Two Dimensions; Vectors Is the Angry Bird going to do any damage? Explain how you came to your conclusion.

Kinematics in Two Dimensions; Vectors

3-2 Addition of Vectors – Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

3-2 Addition of Vectors – Graphical Methods If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

3-2 Addition of Vectors – Graphical Methods Even if the vectors are not at right angles, they can be added graphically by using the “tail-to-tip” method.

ConcepTest 3.1a Vectors I 1) same magnitude, but can be in any direction 2) same magnitude, but must be in the same direction 3) different magnitudes, but must be in the same direction 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B?

ConcepTest 3.1a Vectors I 1) same magnitude, but can be in any direction 2) same magnitude, but must be in the same direction 3) different magnitudes, but must be in the same direction 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B? The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other, in order for the sum to come out to zero. You can prove this with the tip-to-tail method.

3-4 Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other. Left off here in 1st hr

3-4 Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions.

ConcepTest 3.3 Vector Addition You are adding vectors of length 20 and 40 units. What is the only possible resultant magnitude that you can obtain out of the following choices? 1) 0 2) 18 3) 37 4) 64 5) 100 [CORRECT 5 ANSWER]

ConcepTest 3.3 Vector Addition You are adding vectors of length 20 and 40 units. What is the only possible resultant magnitude that you can obtain out of the following choices? 1) 0 2) 18 3) 37 4) 64 5) 100 The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anything in between is also possible, for angles between 0° and 180°. [CORRECT 5 ANSWER]

3-5 Projectile Motion A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

3-5 Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately.

3-5 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g.

A tiger leaps horizontally from a 6. 5-m- high rock with a speed of 3 A tiger leaps horizontally from a 6.5-m- high rock with a speed of 3.5 m/s. How far from the base of the rock will he land? Your turn:

Solution Choose downward to be the positive y direction Origin where tiger leaps from rock Calculate time for tiger to reach ground Horizontal displacement calculated from constant horizontal velocity Solution

Next, determined the UNKNOWN: x displacement. A tiger leaps horizontally from a 6.5-m-high rock with a speed of 3.5 m/s. How far from the base of the rock will he land? The first thing I did was draw this diagram to check my understanding of the problem. Next, figured out what I was GIVEN: x velocity, height, accel. due to gravity Next, determined the UNKNOWN: x displacement. Decided on an EQUATION to find the x displacement. I could use the 2nd equation for x displacement if I knew time. I could not use the 3rd at all because x acceleration is 0, which would eliminate the x displacement component. I have to interpret the y motion in order to find the time. I can do that because I have the height of the rock. Initial x velocity = 3.5 m/s Height = 6.5 m x displacement = ?

Next, I began SUBSTITUTING. Y component first! A tiger leaps horizontally from a 6.5-m-high rock with a speed of 3.5 m/s. How far from the base of the rock will he land? Next, I began SUBSTITUTING. Y component first! y final – y initial = v initial x time + ½ accel.gravity x time squared y final – y initial = y displacement = heigth = 6.5 m v initial in the y direction is 0. So. 6.5 m = 0 + ½ (9.81) x t squared. time = square root of 6.5 divided by 4.9 Time = 1.15 seconds. Now consider the x component of movement. Remember that there is no acceleration in the x direction. So, x final – x initial = v initial x time + ½ acceleratoion x time squared x final – x initial = x displacement , which is what we are looking for. Since acceleration is x direction is zero, then x displacement = v initial x time So x displacement = 3.5 m/s x 1.15 seconds. The tiger lands at 4.025 m from the foot of the rock.

Vector Practice Question of the day: A spy stands on the sidewalk by an apartment building examining a foreign embassy across the street. The spy then climbs 30 m to the top the building and then slides 40 m along a wire at a 20° angle below right to reach a window on the embassy. What is their total displacement from the sidewalk to the embassy window?

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.

Projectile Motion Practice An angry bird is launched from a slingshot on the ground at a speed of 6.8 m/s. At what angle(s) should the sling shot point in order that the bird land on a piggy 2.0 m away?

An angry bird is launched from a slingshot on the ground at a speed of 6.8 m/s. At what angle(s) should the sling shot point in order that the bird land on a piggy 2.0 m away? The first thing I did was draw this diagram to check my understanding of the problem. Next, I figured out what I was GIVEN: total x displacement, resultant v initial speed. TWO IMPORTANT NOTES: We can interpret the bird’s total movement as being two combined halves. The amount of time the bird is travelling toward its maximum height, and the amount of time it is falling from its max height. The motion in these two frames is identical in magnitude, and identical in x direction, but opposite in y direction: The bird is flying up in one half and falling in the other half. But it is flying to right in both halves. But magnitudes are equal, so if the total x displacement is 2 meters, then we can infer that the bird travels 1 meter of x displacement in each half. 6.8 m/s is the angular resultant velocity. This problem is different from falling off the table which we assumed to parallel to the ground. In this case the 6.8 m/s results from the vector combination of the initial y velocity and the x initial velocity. We would have to use tr X O ? Total x displacement is 2 m Resultant v initial speed is 6.8 m/s

QotD The story: enemy forces are minutes away from blowing up your base with a nuclear reactor. They have placed the reactor a mere 30 m from your base. Your cannon is badly beaten up though and can only shoot horizontally. You have one shot to take out the reactor and save the world from evil… a.) Cannon position is 12 m high, what velocity will you use? b.) The velocity of your cannon is 26 m/s, what height should you shoot from?

ConcepTest 3.6a Dropping the Ball I (1) the “dropped” ball (2) the “fired” ball (3) they both hit at the same time (4) it depends on how hard the ball was fired (5) it depends on the initial height From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first? [CORRECT 5 ANSWER]

ConcepTest 3.6a Dropping the Ball I (1) the “dropped” ball (2) the “fired” ball (3) they both hit at the same time (4) it depends on how hard the ball was fired (5) it depends on the initial height From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first? Both of the balls are falling vertically under the influence of gravity. They both fall from the same height. Therefore, they will hit the ground at the same time. The fact that one is moving horizontally is irrelevant – remember that the x and y motions are completely independent !! [CORRECT 5 ANSWER]

ConcepTest 3.6b Dropping the Ball II 1) the “dropped” ball 2) the “fired” ball 3) neither – they both have the same velocity on impact 4) it depends on how hard the ball was thrown In the previous problem, which ball has the greater velocity at ground level? [CORRECT 5 ANSWER]

ConcepTest 3.6b Dropping the Ball II 1) the “dropped” ball 2) the “fired” ball 3) neither – they both have the same velocity on impact 4) it depends on how hard the ball was thrown In the previous problem, which ball has the greater velocity at ground level? Both balls have the same vertical velocity when they hit the ground (since they are both acted on by gravity for the same time). However, the “fired” ball also has a horizontal velocity. When you add the two components vectorially, the “fired” ball has a larger net velocity when it hits the ground. [CORRECT 5 ANSWER]

Angry Bird Investigation You have been given the responsibility to launch an angry bird from a cannon at a given target that is 16.85 m away: a.) find the maximum range of the angry bird given that its maximum initial velocity is 15.0 m/s b.) the angry bird's maximum initial velocity is 15 m/s; what angle(s) is needed in order to hit the target c.) the slingshot is broken and can only fire at a 60 degree angle; what initial velocity is needed in order to hit the target d.) what is the maximum height that the angry bird can reach given that its maximum initial velocity is 15 m/s Angry Bird Investigation

You’ve heard the claim that a bullet shot horizontally will strike the ground at the same time as a bullet dropped from the same height. The typical gun shoots at 896 m/s and the average human is 1.8 m tall. At what time does the bullet land for the gun shot compared to that of the bullet dropped? What is the final velocity of the bullet right before it hits the ground? What is the furthest distance at which the gun can be shot? (taking into account a change in angle) Bullet Shot QotD