Principles of 2D Motion & An Introduction to Projectiles

Vectors and Motion in Two Dimensions Already used vector addition to analyze systems where displacements, velocities, and forces act in non-linear relations to each other. Goal: – To apply the principles of vectors to the analysis of complex 2D systems including General 2D motion Projectiles Combinations of forces in and not in equilibrium Motion along an incline

The bullet problem: Consider the following scenario: – A person holds a bullet in their left hand and a gun designed to fire that same bullet in their right. They fire the gun forward, that is, in the horizontal direction, and they drop the bullet in their other hand at the exact same moment. – What happens? – Which bullet strikes the ground first? – In what location(s)?

Consider an object moving in both the x- and y-direction with a constant speed: VxVx VyVy It experiences both horizontal and vertical displacements, in this case at a constant rate

Likewise, an object which is accelerated uniformly in both directions can move with either increasing or decreasing speed (depending on the sign of the acceleration) along a 2D line. V ox V oy In this case it experiences horizontal and vertical displacements which “increase in equal increases in equal amounts of time.” Remember that description!

What if an object only sees acceleration in one dimension, but not in other other? What does this imply? It would mean that motion in one direction was ___________?___________, while motion parallel to the second axis was _________?_________, that is either speeding up or slowing down in THAT direction only. Answers: Constant velocity Accelerated

Constant velocity in one direction, but accelerated in the other…let’s try constant velocity in the positive y-direction, but accelerating in the positive x-direction. V ox V oy

V ox V oy Same  y Consistently increasing  x

Vectors that are at right angles to each other might be said to be “independent” of each other. This means simply that due to the ninety-degree orientation between them, they share no similar components in Cartesian geometry. As a result, the magnitude of one is unaffected by the magnitude of the other and vice versa. A nice summary of the prior slide might be to say simply that the horizontal (x) acceleration does not affect the vertical (y) speed. Going back to the bullet problem Let’s examine that scenario. What makes the bullet dropped by hand fall? What makes the bullet fired from the gun fall? Is there any difference? Does the fact that the bullet in the gun is fired forward (x) have any impact on the acceleration of gravity in the y-direction? Does gravity change because the gun’s bullet is moving forward? Gravity Nope, it’s the same planet under both bullets that pulls them down. Again, no. These vectors are independent. Certainly not! If you jump straight up doesn’t gravity work on you the same as if you run forward and jump a the same time? Just because the bullet is moving VERY fast horizontally does not negate the vertical effect of gravity on it.

Still having difficulty wrapping your head around it? Let the Mythbusters help: Here’s a great video of that exact scenario, with a tiny bit of experimental error (0.039s worth). Interesting to see that the “dropped” bullet actually hits first due to this experimental error, which puts to rest any further misconception.

Here’s another take: Same distance fallen in each frame Galileo’s hypothesis of gravitational acceleration: Equal increases in distance in equal amounts of time. Was Galileo correct? Let’s focus on the y-direction first. Photo Source: Holt, Physics

Now let’s focus on the x-direction. Red ball has no X-velocity. Observe the horizontal distance traveled by the yellow ball in between each frame. It’s approximately the same horizontal distance each time. What does this tell us about the horizontal motion of the projectile? If you answered that it is moving sideways with a constant velocity, you get a prize.

Conclusions: 1)Vectors at right angles are said to be independent of each other. 2) A vertical force/acceleration does not affect a horizontal velocity and vice versa. 3) The vertical force of gravity has no effect on the sideways motion of a projectile. 4) Even an extreme example of conclusion number 3, like the bullet problem with real life considerations like air resistance, demonstrates the validity of that claim. Can you think of any exceptions and why they may not follow? What if the objects are not the same, like a hammer and a feather? Hard to believe but gravity still affects each the same, but since we have different shapes in this example the problem of air resistance is exacerbated. Do a web search for “Apollo 15 hammer feather”