These notes are seen pages. A quick summary: Projectile motion is simply horizontal motion at constant elocity with ertical motion at constant acceleration. An object moing in a circular path experiences centripetal (i.e. toward the center ) acceleration; which is related to the speed of the object and the radius of the circle. The elocity of an object in a moing reference frame is the sum of the elocity of the object within that reference frame and the elocity of the reference frame itself. Projectile Motion A projectile is any object in freefall, i.e. an object in the air and without any means to control its own motion. In other words, a projectile is completely at the mercy of graity. We can consider a rock, a baseball, a bullet, etc to be a projectile when airborne. The study of projectile motion is important: we moe many things by throwing (or launching, or shooting) them into the air. Fortunately, since we hae already studied motion in one dimension, the study of projectile motion is fairly straightforward. A key principle for projectile motion is that horizontal and ertical motion are independent of each other; they are only connected by time. That is, elocity in the horizontal dimension only affects the horizontal position of an object. And horizontal acceleration only changes the horizontal elocity of an object. The same is true with ertical elocity, position and acceleration. For this reason, we can analyze the motion of a projectile as two independent motions: motion in one dimension in each of the horizontal and ertical dimensions. We will approach projectile motion problems in exactly the same way we approached motion in one dimension problems. But projectile motion problems are simpler, in a way. In Chapter 2, we identified points of interest in the motion of an object. These points defined interals of constant acceleration, and some problems had as many as four points with three different interals. An object in projectile motion has only one acceleration: that of graity. This acceleration only acts in the ertical dimension; the horizontal acceleration is zero. Since all of projectile motion takes place with constant acceleration, there is only one interal of motion. The rules for projectile motion problems are ery similar to the rules for Motion in One Dimension problems from Chapter 2: 1. Draw a picture 2. Define x and y axes and label the origin 3. Label the coordinates of your initial and final points; use letters only, no numbers! 4. Make a table of the information for x and y dimensions 5. Write the final position equation for each dimension 6. Count your equations and unknowns... and then sole! Page 1 of 7
For most projectile motion problems, the only necessary equation is the final position equation. Using this for each dimension gies you two equations and typical problems leae you with two unknowns. On occasion, you may hae to also use the final elocity equation. Note: If a projectile motion asks for or gies you any information related to either the final elocity in the x direction or the final elocity in the y direction, you must also write the final elocity equation for each dimension! If you do so, they will look like this: fx = ix + a x t fy = iy + a y t All of the information on the right side of these equations can be found in your table. Circular Motion (Centripetal Acceleration) One of Isaac Newton s great insights into the nature of motion, and something that confounded Galileo, was that objects naturally will moe in a straight line at constant speed, i.e. with constant elocity, unless acted on by a force. His next logical discoery was that forces cause objects to accelerate. We will inestigate Newton s work in Chapter 4 and see how Newton s Laws of Motion define the relationship between force and acceleration. But for now we are interested in just recognizing acceleration when it occurs. Newton suggested that any change in elocity should be defined as acceleration. This is fairly intuitie if we think about acceleration as increasing speed. Howeer, decreasing speed also qualifies as acceleration (we did this in Chapter 2... acceleration is a ector and an acceleration that is opposite the direction of motion, e.g. if elocity is in the positie direction and acceleration is in the negatie direction, will result in a decrease in the speed of the object). Newton realized that a third possibility exists: if an acceleration acts perpendicular to the direction of motion, the object will neither increase nor decrease its speed. But its elocity must change. Therefore, the only possibility is that it changes the direction of its elocity! The consequence of the definition of acceleration and how it relates to elocity is simple: Acceleration in same direction as elocity: Acceleration opposite to direction of elocity: Acceleration perpendicular to direction of elocity: speed increases speed decreases object changes direction Newton took this concept to the next logical step: if an object experiences a continuous, constant acceleration perpendicular to its direction of motion, the object will experience a continuous, constant change in direction. In other words, it will turn in a circle! Page 2 of 7
From this discoery we make a definition: centripetal acceleration. The word centripetal means center seeking. We will use this concept repeatedly throughout the rest of the year. Centripetal acceleration will always be associated with circular motion. The association is simple: Any object that moes in a circular path is accelerating. The acceleration associated with the circular motion is called centripetal acceleration because the direction of the acceleration is directly toward the center of the circle. Note that the direction of the acceleration must be toward the center of the circle because as the object moes in a direction that is always tangent to the circle, the acceleration is always perpendicular to its path. Basic geometry dictates that a radius of a circle is always perpendicular to its corresponding tangent (i.e. the radius and tangent drawn at the same point on the circle will be perpendicular to each other). Sometimes we will call centripetal acceleration by the name radial acceleration. They mean the same thing and are always associated with circular motion. Acceleration in a straight line is ery easy to calculate: the change in elocity is simply the change in speed. So you can take the difference between the final and initial speeds and diide by the time interal. But how can we determine centripetal acceleration, when there is no change in speed? This was the question that Isaac Newton thought about for a long time... and which led him to inent calculus and discoer uniersal graitation. To answer this question, Newton drew a picture (of course... what else?) that looked similar to this: He then considered that while the speed of the moing object was constant, so the magnitudes of the initial and final elocities (i.e. the length of the arrows) were the same, the directions were different. This allowed him to define the change in elocity... not by considering the difference in speed, but by subtracting the ectors themseles! Page 3 of 7
How do we subtract ectors? Exactly the same way we subtract numbers... we add the negatie. The negatie of a ector is the identical ector that points in the opposite direction (note: remember that we use plus or minus signs for direction when we define our motion ectors; because of this definition, it should make sense that the negatie of a ector is the identical ector pointing the opposite direction.) Consider the following: We can start with ector A and add ector B, or we can subtract ector B. To subtract ector B, we find -B and add it to ector A. The result is the difference between A and B, or ector A B. Algebraically, A minus B is the same thing as A plus -B. Vectors work on the same principle. Newton was able to do this with his elocity ectors. He subtracted them, graphically, and created a triangle with sides i f and, where is simply f i. Page 4 of 7
He recognized that this triangle is an isosceles triangle, since i and f hae the same magnitude. He then created another triangle within his circle with the two radii drawn to the initial and final positions: With the use of simple geometry, it is easy to show that the angles θ and φ are identical (note that the angle each radius makes with its elocity ector is 90 degrees)... so the two triangles are similar triangles! (This is because they are both isosceles triangles with the same angle between the congruent sides.) Using another useful tool of geometry, that the ratios of corresponding sides of similar triangles are equal, Newton was able to show: = d With the use of calculus, he was able to show that as the angle θ is made smaller, i.e. the initial and final points are brought closer together, the distance d becomes the same as the actual distance the object traels around the edge of the circle. The distance between the initial and final point can be written in terms of speed and time (i.e. speed multiplied by time): = t (note that is just the speed of the object as it moes in a circle) And it was now that Newton made a simple but important discoery by rearranging this expression: t = 2 Page 5 of 7
For an object in circular motion, the magnitude of its change in elocity with respect to time is gien by simply 2 /, where is the speed of the object and is the radius of its circular path. Newton noticed (and we should notice!) that this is the definition of acceleration! Therefore we hae: magnitude of centripetal acceleration: 2 a c = Calculating centripetal acceleration is ery simple: you just need to know the speed of the object and the radius of its path. Notice that if speed is in m/s and is in meters, the units of a c are m/s 2, just as they should be. Alternatiely, you might be gien the acceleration and the radius of the path and asked to calculate the speed of the object. Or you could be gien a and and asked to calculate. In any case, this expression simply shows us that three quantities (centripetal acceleration, speed of the object, radius of the turn) are connected to each other; gien any two, we can easily calculate the third. elatie Velocity elatie elocity is a fairly simple idea: if you are walking inside a train at a speed of 3 mph, and the train is moing at a speed of 25 mph, then objects outside the window will moe past you at a speed of 28 mph. In other words, your speed relatie to the ground is your speed relatie to the train plus the train s speed relatie to the ground. Galileo was the first to formalize relatie elocity in his books on the study of motion. He took the simple idea aboe and adapted it to include not only motion in one direction, but motion in any direction. The adaptation was simple: An object s elocity relatie to the ground is the object s elocity within its frame of reference plus the elocity of the frame of reference relatie to the ground. We will write this idea as an equation by using: 1 2 3 is the object s elocity within its frame of reference is the elocity of the frame of reference relatie to the ground is the elocity of the object relatie to the ground With these definitions, we can write: 1 + 2 = 3 Page 6 of 7
In the simple example of you walking in the train, the train is your frame of reference. Imagine the train s windows were closed so you could only see the inside of the train. You can only measure your speed relatie to the train. You can only use the train as reference for your motion. So we call the train your frame of reference because it is what you are measuring your speed relatie to. Notice that if 1 is your 3 mph and 2 is the train s 25 mph, then 3 is 28 mph and represents how fast you moe relatie to the ground outside. The train is a simple example for a reason: we always try to start with simple examples before we expand to more general examples. Fortunately the more general examples are not much more complicated. The two problems I hae gien you for homework are two dimensional, so we adapt the equation aboe for two dimensions: x direction: y direction: 1x + 2x = 3x 1y + 2y = 3y To sole relatie elocity problems, you must: Draw a picture (of course) Identify the object and its frame of reference Identify the elocity (speed and direction) of the object in its reference frame Identify the elocity of the reference frame relatie to the ground Identify the elocity of the object relatie to the ground Note that the elocities might hae to be split into x and y components. You then must consider the relatie elocity along the x and y dimensions separately. Keep in mind that there are three elocities to work with in each problem; you will typically be gien two and asked to find the third. Page 7 of 7