Lecture 6: 2-Dimensional Motion and Projectile Motion

Displacement, Velocity, Acceleration
The displacement vector is represented by
Dr = r2 - r1
Therefore, the average velocity is represented by
v = (Dr/ Dt)= (r2 - r1)/(t2 - t1)
And acceleration can be represented by
a = (Dv/ Dt)= (v2 - v1)/(t2 - t1)
Relative Velocity
                    A coordinate system, S', moves with a constant velocity u .  Another coordinate system, S, is at rest relative to S'.  An event P occurs in the frame S',  the figure below illustrates the various relationships between the position vectors  of the event P in both reference frames.

r = r' + ut  or r' = r - ut

If we take the derivative of the position vector, we will get the velocity vector
v' = v - u
If we take the derivative of the velocity vector, we get the acceleration vector
a' = a
Therefore, since u is constant, the acceleration is the same in all reference frames.

PROJECTILE MOTION
A. Introduction
1) In this section, we look at the motion of objects as they move in a two dimensional vertical plane in the presence of gravity. Such an object is called a projectile (as in to project or launch an object).

2) Since this is motion in two-dimensions, we look at the motion in both the x-direction and y-direction. We also assume that the only acceleration being experienced by the object is the acceleration due to gravity (9.80 m/s2, directed downward toward the center of the Earth).

B. Properties of Projectile Motion
1) Projectile motion is a combination of un-accelerated x-direction motion and free-fall.

2) Since there is no acceleration in the x-direction, the velocity in the x-direction is constant.

3) The maximum horizontal distance reached by the projectile is called the RANGE.

4) The maximum vertical distance reached by the projectile is called the MAXIMUM HEIGHT.

5) The velocity of the projectile in the y-direction is equal to zero at the maximum height reached by the projectile (just like the motion of an object in free-fall, the object thrown upward has to stop before it eventually comes back down).

6) The vertical and horizontal motion of a projectile is independent. To illustrate: let’s say that I have 2 golf balls on a table. The first golf ball is pushed off of the table at the same time the second one is dropped from the table. The ball that was pushed undergoes some horizontal motion yet they both reach the ground at the same time (see figure below).  This is because the vertical motions of each ball is identical.

7) The total time it takes for a projectile to hit its target has a lot to do with how high the projectile has to go. For example, target 1 is 30 m away and target 2 is 50 m away. If I launched two identical arrows with identical initial speeds toward the two targets with the arrow aimed at target 1 reaching a maximum height of 50 m, and the arrow aimed at target 2 reaching a maximum height of 30 m. Which target will be hit first?

            ANSWER: The most common wrong answer is that target 1 will be hit first because it is closer. However, the arrow has to go much higher than arrow 2. The correct answer is target 2 will be hit first because the arrow does not have to go as high as arrow one. Think of it this way, which arrow will hit the ground first, on that is dropped from 50 m or one dropped from 30 m? The reason for choosing the 30-m answer is the same reason why target 2 is hit first.

8) Finally, the path traced out by a projectile is a parabolic path.

C. Equations of Projectile Motion