Physics
Chapter 3: 5-8 Displacement and Velocity
Web Lecture
The Mathematics of Kinematics
Unit Vectors
Introduction
Galileo Galilei is credited with a number of key insights into mechanics, astronomy, and the methods of experimentation. His work with "falling bodies" provides us with the first classic set of repeatable experiments. This study of motion was in part a response to the desire for a better understanding of ballistics -- the motion of projectiles -- that was fundament to the new military tool called the cannon. To slow down the speed of the falling object so that he could time it using the most accurate instrument available at the time, a water clock, Galileo sent balls rolling down an inclined plane. His discussion on this uniform difform motion bears a striking resemblence to that of Nicole Oresme two hundred years earlier.
Galileo realized that the only force acting on the projectile was the force of gravity, which caused constant acceleration of the projectiles. The distance a body fell was proportional to the time squared, and was independent of any other factors, including the mass of the object.
Outline
Unit Vectors
Unit vector Notation
From the foregoing discussion, you should notice that Dx and Dy are just vectors of D magnitude in the x and y directions. Many texts use a notation that separates the magnitude and the direction components of the vector for clarity. In this notation, the unit vectors i, j, and k, correspond to vectors of unit 1 length (whatever scale or unit is being used) and the directions x, y, and z respectively. In this notation, our Dx and Dy vectors become xi and yj. (John is not yet ascending skyward...wait for the section on projectile motion!). We can pull out all the D magnitudes from the first three legs of the trip, and do a sum that looks like this:
(Dx1 + Dx2+ Dx3)i + (Dy1 + Dy2+ Dy3)j = D
Relative velocity
So far, we have used a fixed frame of reference: a coordinate system based on John's home. We chose that for the sake of convenience (and we are correct in doing so). But that reference system is not the only one we could use. We could have placed our coordinates' origin at the library, with y postive to the west, changing both origin and orientation. We could also have placed the frame of reference on John himself, so that his displacement was always zero, but the library, Main Street, and home changed displacements as he moved.
Lest a moving frame of reference seem too farfetched, a little reflection should show you that we actually do this all the time. Think about the last time you ran errands and the gas tank got low. The question "how far am I from the nearest gas station" assumes a point of origin based on your car's current location. The next time you get low (and ask the same question about the same gas station), you probably won't be at the same intersection--so the distance to the nearest station and the direction you must go to get there will change. But your car will still be your point of origin for determining those factors.
In comparing velocities between two moving objects, we often need to see one object's velocity from the frame of reference of the other. Consider a blue, 1969 Volvo Sedan, facing southward on the boulevard. but stopped to make a turn into an alley. Approaching from the rear is a tan Toyota Tercell, speeding at 55 mph, and much too close to stop in time to avoid colliding with the Volvo. The Tercell driver applies the brakes, but will still hit the Volvo at 40mph. The driver of the Volvo notices the Tercell in the rear view mirror, puts the car in gear, and steps on the gas. The Volvo is moving at 10 mph when the Tercell hits it, so the collision is 40mph - 10mph = 30mph. While enough to total both (old) cars, this is not enough to injure any passengers properly seat-belted. [Yes we all walked away from this accident, and yes, I actually did think if I can get our car moving I will reduce the overall energy of the collision on a mile-for-mile basis--because going 5 mph reduces a 40 mph collision to a 35 mph collision while putting the car in gear, thus demonstrating the practical use of physics in everyday life].
We can determine the difference in velocities at the time of the collision graphically or by analysis. Either way, we should use the viewpoint of one of the cars in order to determine the appropriate sign to apply. From the point of view of the stationary Volvo, the Tercell is decreasing distance, and has a velocity vector in the negative direction, of -40mph; from the moving Volvo several seconds later, the Tercell now has a -30 mph velocity, still destructive but much less dangerous to the passengers. Be sure that you investigate the frame-of-reference possibilities when you are working the Interactive Physics exercises.
Vector kinematics
We can now generalize all those formulas from last chapter in terms of their vector components.
The instantaneous change in V is the limit of the individual vector changes. If we can find the instantaneous velocity in each component direction, we can use vector addition to find the overall instantaneous velocity of the object.
Projectile motion
Now we can look at another example of two (or three) dimension motion: throwing things into the air. As mentioned earlier, this was the problem that interested Galileo and his contemporaries, who wanted to understand how to accurately aim their canon at neighboring but offensive Italian city-states. We try to sublimate our desire to throw things by aiming spacecraft at neighboring but remote planets (see the latest on our efforts at www.nasa.gov).
Simple projectiles launched from the earth's surface are also useful because they undergo constant gravitational acceleration, a phenomenon which is readily observed, and so experiments are easily repeated, everywhere on earth (at the same altitude). If we orient our coordinate system in the direction of the horizontal motion, and there are no sideways forces (wind, jets) acting on the projectile, then we can break down projectile motion into a problem of two dimensions: horizontal and vertical. The horizontal velocity remains constant and we only have to worry about the effect of gravitational acceleration on the vertical velocity vector.
Note that the two velocity vectors are independent of each other: if something affects one vector, it doesn't affect the other. A case in point is one of your lab exercises, which involves launching a marble horizontally, while at the same time dropping a second marble from the same height with no horizontal motion--your goal is to observe and explain what happens.
Baseball, what else?
Wiki Commons
Ah, the sound of the ball hitting the sweet spot on the bat, and then the sight of that perfect parabola arching out toward the 405 foot marker on Safeco'sT-Mobile center field fence, and the small white speck of the ball dropping BEHIND it....If only a Mariner had hit it instead of a Yankee!
Since there is no way the Mariners can make it to the playoffs this season*While we wait in suspense to se whether the Mariners will actually have a shot at post-season play, we Mariners fans have to find something else to do at the ball game. The last time I went to a game (tragic loss in the 12th -- at least we made them work for it), my seats were on the general admission level along the first base line, at least 5 stories up (given the buildings next to Safeco that I could see over). Let's estimate the height as 35m (114 feet). The home run hit by the nameless opposition in the top of the 12th inning cleared the 405 foot mark with a few inches to spare, and it was more or less at my eye-level at the top of its arc. So how fast was it going when it left the bat?
We do have to recognize that the ball has initial velocities in TWO directions: up and centerfield-ward. Its up/down motion is controlled by the earth's gravitational field. Its outward-bound motion will be affected by wind resistance, but let's ignore that for the moment, [although in fact the batters can't -- at Safeco, the incoming breeze from Elliot Bay tends to drive balls toward the first base line, so you have to hit a bit left of where you want the ball to go]. With these assumptions, we can analyze the flight of the ball. The batter took a pitch that crossed homeplate about waist-high, so let's say the starting height of the ball is 1m.
First: the ball is hit with an upward velocity that allows it to reach 35m in height before falling back down. At the top of its arc, the vertical velocity of the ball is zero. The only force acting on the ball is gravity, which is accelerating downward. Let's assume that up is positive and down is negative, which means gravitational acceleration is -9.8 m/s2. So we can use v2 = v02 + 2a(x - x2) to figure out the initial vertical velocity of the ball:
| We need v0, so we isolate it first: v2 - 2a(x - x0) = v02 |
| 0 - 2 * (-9.8 m/s) * (35m - 1m) = v02 |
| 0 - (-19.8m/s) * (34m) = v02 |
| 0 - (-673.2) = +673.2 = v02 |
| √673.2 = 25.9m/s = v0 |
Since the ball falls (v0 = 0 ) from 35 to the ground, we can determine its time in the air from y = ½ g t2.
| s = ½ g t2 |
| 35m = ½ (9.8m/s2) t2 |
| 35m = (4.9m/s2) t2 |
| 35m / (4.9m/s2) = t2 |
| 7.4s2 = t2 |
| 2.67s = t |
The time it took it to rise 34m will be the time it took it to fall 34m: 34m/4.9m/s2 = t2, so 2.63 seconds to go up. The ball was in the air 2.67 + 2.63 = 5.3 seconds.
During this time, the ball travelled 405 feet to the fence, or 123.4 meters. Its horizontal velocity would be 123.4m/5.3 sec = 23.3m/s.
We can now figure its net velocity from the knowledge that it is going 23.3m/s in the x (horizontal) direction and 25.9 m/s in the y or vertical direction. Net v2 = vx2 + vy2 = (25.9 m/s)2 + (23.3 m/s)2 = 670.81 + 542.1 = 1212.91. v = 34.83 m/s (or about 78 mph), slower than a line drive but fast enough to be a home run.
The angle the ball makes with the ground is the tangent of y over x: tan-1 23.3/25.9 = tan-1 .899 = 37°.
Practice with the Concepts
Discussion Points
- If you stand holding a large umbrella in a windless rainstorm (the drops fall vertically), will you get wet? What happens if you start running through the rain, still staying directly under the umbrella?
- Can more than one trajectory land a projectile at the same target point?
*This particular line has, unfortunately, not been updated since 2001.
© 2005 - 2024 This course is offered through Scholars Online, a non-profit organization supporting classical Christian education through online courses. Permission to copy course content (lessons and labs) for personal study is granted to students currently or formerly enrolled in the course through Scholars Online. Reproduction for any other purpose, without the express written consent of the author, is prohibited.