chapter3

Chapter 3 Motion in 2 and 3 Dimensions

The biggest difference between one-dimensional and multi-dimensional motion is the use of **vectors. Vectors are quantities with magnitude and __direction.__ ** ﻿ Examples of vectors are velocity, displacement, and force. Vectors are equal if they have equal magnitude __and__ direction. Vectors are made up of components. Components are two or more vectors that add up to one vector. Generally, the components are in the x and y directions. The angle  θ formed between the positive x-axis and the vector can be used to calculate the components, with the following relationships:

We sometimes express components in terms of unit vectors, usually i (x-direction), j (y-direction), and k (z-direction). These vectors all have a length of one unit in their direction. To write vectors with unit vectors, we multiply the unit vectors by scalars and add them. For example, if a vector has an x component that is 6 units long and a y component that is 2 units long, we could write it as 6i+2j. There are several operations that we can perform on vectors.

Addition: The head-to-tail method of adding vectors places the vectors end to end, with the beginning of one in contact with the tail of the other. Once all the vectors are thus connected, the resultant vector is drawn from the start of the first vector to the end of the final vector:

The second method of addition is the parallelogram method. It only works with two vectors. The two vectors begin from the same position, and then a parallogram is formed. The resultant vector goes from the original vertex to the diagonal vertex:

To add vectors non-graphically, simply add together their components. For example, to add the vector 5i+2j and the vector 13i-7j, add together the i numbers and the j numbers to get 18i-5j. Multiplication: Vectors can easily be multiplied by scalars. Simple multiply the magnitude of the vector by the scalar, which is the same as adding together vectors (multiplying by two means adding two vectors, multiplying by 3.456 means adding 3 and .456 of a vector). When multiplying by a negative number, simply reverse the vector's direction, then multiply. For example, to multiply a vector by -4, first flip it 180 degrees, then multiply the new vector by 4.

The most pertinent vectors for motion are position (r), velocity (v), and acceleration (a) vectors. Recall from one-dimensional kinematics that these quantities are related by calculus:

Projectiles

In projectile motion, the horizontal and vertical velocities are independent. This means that an object launched straight horizontally (positive horizontal velocity) will take the same time to fall as one that is simply dropped (zero horizontal velocity). The following clip from Mythbusters shows this principle:

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Sample projectile problem (if you can do all parts of this problem, you know all you need to about projectiles):

An arrow is launched at angle θ and velocity V0 from ground level (h=0). The arrow is fired from and travels over a flat surface. Give all answers in terms of V0, g, and θ.

a. At what time will the arrow reach its maximum height? b. At what angle should the arrow be launched to achieve the maximum vertical displacement? c. At what angle should the arrow be launched to achieve the maximum horizontal displacement? Solution below

What about projectiles hitting unusual surfaces, such as a golf ball hitting a hill? The way to approach these problems is to write the three equations you know (the x-distance, y-distance, and equation of the surface) and then solve the resulting system. For example, say you have a golf ball hit from the origin onto the hill described by y=1.2x with initial velocity v at angle Ɵ. Set up the following system: y=1.2x x=v * cos Ɵ * t y=v * sin Ɵ * t - .5gt 2

You should be able to solve this system. If you can not, then AP Physics may not be the right class for you.

An interesting article about ENIAC, the first computer, which was used to compute trajectories of projectiles during World War II, can be found [|here]

Circular motion Circular motion, for now, is not very hard. When objects move in a circle, they must accelerate towards the center in order to maintain a circular path. This is known as centripetal acceleration, and is related to the velocity and radius by the equation: v 2 /r = a c where v is the velocity, r is the radius of the circle, and a c is the centripetal acceleration. This formula says that a greater velocity causes a greater acceleration, while a greater radius causes a smaller acceleration. A proof for this formula can be found here: []

Note that an object in circular motion may also have a tangential acceleration in addition to the centripetal acceleration. In this case, the total acceleration is the sum of the two acceleration vectors.

Real-life examples of circular motion include a car going around a turn, a centrifuge, and a [|particle accelerator]

This picture sums up what you need to know about circular motion:




 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
 * Content || 3 || 15 ||
 * Organization || 4 || 20 ||
 * Accuracy || 4 || 20 ||
 * Appearance || 4 || 20 ||
 * Participation || 3 || 15 ||
 * TOTAL || 90 ||