It is really important to know how to use vectors for the rest of the class. Vectors as we said before have both magnitude and direction. Commonly this is going to be force and acceleration, but there are others.
When using Vectors, make sure to use components together. Life before with Vector A and Vector B in order to add them together or subtract them from each other. We will have to take components of both vectors and do the arithmetic in that manner.
In 2-D motion, vectors can be broken down into horizontal and vertical motions. Horizontal motion will be broken into the x component and Vertical broken into y components. The Math of vectors depends on the components being kept together in the same directions.
How do we actually break vectors into components? The answer lays in the trig fun sin, cos, and tan. If you don’t know how to use these functions make sure to brush up on the subject before you go further. In the terms of vectors the vertical direction will be the y component due to the change on the y axis. And the horizontal will be the x component due to the change in the x axis.
So using the trig function for the components we can find these values. Vx= V cos (theta) for the x component. Vy= V sin (theta) for the y component. This will come in handy in this chapter and the next.
How do you find the magnitude and angle of the vector? If you are given a vector, you can find the components of the vector in x and y components. If you are just given the components you can use the Pythagorean theorem to find the total vector magnitude, or you can use the tan function. By using the invert tangent, you can find the angel of a vector.
Keep in mind the direction for vectors on a Cartesian coordinate system implied the sign. Moving in the positive direction implies positive numbers. where as movement in the negative direction implies negative numbers.
Acceleration is a vector and can be split up into components as well. The x component will be a change in the x direction and the y component will be the change in the y direction.
If we need to find the change on velocity it can be broken down as well into components of a vector.
When learning projectile motion we not only look in the x direction but we also view the y direction and the angle. Fortunately acceleration will be gravity for most cases (-9.8 m/s2 ) in a downwards direction. We will use the same kinematic equations as before however, we will be breaking them into x and y components. For finding initial velocity we can use trig rules to find the components.
For example. Initial velocity of a ball is 5 m/s and is launched at 30 degrees above the horizon. To find the x and y components of the initial velocity we will use trig functions to do so. Vix = Vi cos (theta) and for the y component Viy = Vi sin (theta)
It is important to remember that when gravity is pointing downwards, there is no effect on horizontal motion. This independance of horizontal and vertical motion is very important.
This time a projectile spends in the air is dependant on the vertical displacement and the vertical component of velocity.
In this video, you can see that even when the two have different initial velocities, they both accelerate in the same direction at the same rate due to gravity in the vertical direction.
The equations of motion for constant acceleration are still valid for projectile motion. However, the initial velocity now has two components in the horizontal and vertical directions.
For projectiles only under the influence of gravity :
Things to remember.
Becuase the acceleration in the horizontal direction is equal to 0, a projectiles velocity in the horizontal direction is a constant.
The vertical component of the velocity and the vertical displacement are identical to those of a free falling object.
Projectile motion can be described as a superposition of two independent motions.
The range equation is another useful equation that gives the range of a porjectile for a given initial velocity and projection angle. However this only works when the launch and landing points are the same evelvation, other wise you will have to use a different method.