Motion Under Constant Acceleration 2D Motion

We are now beginning a discussion of motion in 2D. For the most part, this motion is analogous to what we have discussed already for motion in 1D. Projectile motion is one of the most common forms of motion in 2D under constant acceleration.

This occurs when you hit a baseball, throw a ball, kick a soccer ball and several other familiar instances. However, even though the object is now moving in 2D, that is it has a change X and Y position with time, it is still governed by the basic 1D equation of kinematic motion that we have been working with.

v(t) = v_o + at

The graphical representation of this is shown below:

To give you an empirical understanding of this, three demos were done in class.

• Demo 1: A cart moving on a frictionless track would shoot a ball vertically upward at a precise time and continue to move at constant velocity.

  Approximately 1/3 of the class thought the ball would go straight up and down and land behind the cart. However, the ball landed in the cart every time.

  What's the source of the misconception?

  To think about this problem its important to isolate the forces in the X and Y directions. As the cart is moving at constant velocity in the X-direction when the ball is fired, there can be no force in the X-direction. Therefore the ball and the cart share the same velocity in the X-direction. As long as there are no forces in the X-direction, the x-position of the ball and the cart must remain the same with time.

  In the animation below, the X-position of the ball and the ball launcher remain the same, always.

  

  Demo #2: Here we consider the case of two balls released simultaneously. Ball 1 has no x-velocity associated with it and therefore falls straight down. Ball 2 does have an x-velocity associated with it and hence moves horizontally as it is fired. The question was whether or not the two balls would hit the ground at the same time? Again, some members of the class though that the horizontal motion of the ball would mean it would hit later than the one that was just dropped.

  Once again the following animation should help. There are no forces in the X-direction so its velocity is constant. Since both balls are subject to the exact same acceleration n the y-direction (due to gravity) then both will hit at the same time. Thus if we drop a cannon ball at the precise time that the other one is fired, both will hit the ground at the same time. Notice that the velocity in the X-direction of the cannon ball remains constant.

  

  Another example of this common shared movement in the X-direction is shown in the animation below. We are of course neglecting air resistance here. Since the package and the plane both have the same X-velocity and this is constant then after the package is released it maintains the same X-position as the airplane. Th package falls towards the earth in a curved path which is specified by a parabola. Notice that the vector in the X-direction maintains a constant length (constant velocity) but the vector in the Y-direction continues to increase as the Y-velocity continues to increase due to constant acceleration in the Y-direction.

  

• Demo #3: Shoot the Garfield. Once again, the dart always hits Garfield because they are both accelerated in the Y-direction the same. The initial kinetic energy of the projectile determines at which Y- position Garfield will be hit. This is shown below for two cases. The top animation the bananas is thrown relatively fast and hits the Monkey shortly after the Monkey falls from the tree. The bottom animation is the case of a slow banana. It still reaches the Monkey but it takes longer and the Monkey as fallen more. Again, there are no forces in the X-direction which is why this works.
  
  

So even though we are now in 2D we are still interested in measuring position and velocity as a function of time. However, in this case we will be plotting the X and Y components of velocity and position.

Here is an example of an X-Y trajectory for some object in motion.

If we examine the specific time interval in the above trajectory, see between 3 and 4 seconds, we can asses the velocity in both the X and Y-directions. This is shown below:

We will be studying projectile motion both in the atrium and in Applet land. For now, the following two graphs summarize the kinds of observations you can make about projectile motion:

Relevant parameters:

• Total horizontal range
• Projectile angle
• Time of flight
• Maximum height of projectile
• Time it takes to get to maximum height
• Initial velocity

All of this will be relevant to ballistic cows on the last day of class.