Fundamentals of Physics/Motion in Two Dimensions

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The Basics[edit]

The goal of this section is to understand how objects move in fully two dimensions. By contrast, one-dimensional motion concentrated on motion strictly along the x or y axis. Two dimensional motion is where an object undergoes motion along the x and y axes "at the same time." The position of an object in two-dimensional space can be plotted by its (x,y) coordinate. These coordinates are found by the equations

x=x_0+v_{0x}\Delta t+\frac{1}{2}a_x\Delta t^2

and

y=y_0+v_{0y}\Delta t+\frac{1}{2}a_y\Delta t^2.

Note that also evolving as an object moves are its speeds along two axes as well via

v_x=v_{0x}+a_x \Delta t

and

v_y=v_{0y}+a_y \Delta t.

x and y components of velocity under free fall

Remember that the x and y coordinates are perpendicular to each other, that is the x and y axes are orthogonal. This is a special relationship in math and physics, and means that processes along one axis do not affect processes along the other axis. Therefore, whatever happens along the x axis does not affect what happens along the y axis, and vice-versa. This is a key concept to understand. Two-dimensional motion is sometimes called "projectile motion" which encompasses objects flying through space under the influence of gravity. Baseballs, cannon balls, basketballs moving through space are all examples of projectile motion. Near the surface of the earth, projectiles in flight are restricted to motion where a_x=0 and a_y=-g=-9.8 m/s2. You can immediately find forms of the x= and y= equations above, given these restrictions. You can also find v_x and v_y equations, noting that v_x will always be equal to a constant (vx0) since by orthogonality, g (gravity) only affects y and vy.

The Need for Vectors[edit]

At any given time, your object will have four quantities describing its motion: x, y, v_x, and v_y. Since position and speed now each have two components (or parts), position and speed will be "vectors," called {\vec r} and {\vec v} respectively. {\vec r} will consist of two components, the x and y coordinates of the object. Similarly, {\vec v} will consist of the components v_x and v_y. As you will now see, the two components of both {\vec r} and {\vec v} gives them both a magnitude (strength, length, etc.) and direction, which you must know how to handle.

Vectors: magnitude and angle[edit]

There are two ways of dealing with vectors, and you should be proficient with both. The first way is in "magnitude-angle form," where you report the magnitude of the vector and the angle at which it is pointing. For the position, the magnitude (or total distance from the origin) is r=\sqrt{x^2+y^2}. The angle this vector will make relative to the +x-axis is given by \theta where \theta=\tan^{-1}\frac{|y|}{|x|}. The absolute value signs are important to remove any negative values that might pop up and ensure the angle is with respect to the +x-axis. The velocity vector is tracked similarly, namely v=\sqrt{v_x^2+v_y^2} with \alpha=\tan^{-1}\frac{|v_y|}{|v_x|}, where \alpha is the angle the velocity vector makes with respect to the +x-axis, and is essentially the direction the object is moving in at that instant of time. Be sure you understand why a vector has a magnitude and an angle, and be sure you can always compute both from a given vector's components.

Vectors: Component form (or i,j,k notation)[edit]

The other way of handling a vector is in "component form." In this form, you list each component directly, next to a unit vector specifying what axis the component goes to. So if an object is 5 meters along the x-axis and 2 m along the y-axis then {\vec r}=5{\hat i}+2{\hat j}, where {\hat i} and {\hat j} are unit vectors meaning x-axis and y-axis, respectively. Likewise, if an object has an x-component of velocity of 3 m/s and a y-component of -2 m/s, its velocity vector would be {\vec v}=3{\hat i}-2{\hat j}.

Circular Motion[edit]

The other type of two-dimensional motion that is important is circular motion, which describes how an object moves in a circle. In this type of motion, the object is always has an acceleration that points toward the center of the circle around which it traveling. If you choose a circle of radius r and want the object to move around the circle with a speed v, then the strength of the acceleration, called the "centripetal acceleration" must be a=\frac{v^2}{r}, and it must always point toward the center of the circle.