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Linear Motion SPECIAL CASES of Linear Motion

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Linear Motion SPECIAL CASES of Linear Motion A.P. Physics (B) Exam “Quick Reference” Study Guide This study guide is to provide you with a source of all the information that I [Mr. Welkley] consider the MOST important for you to have essentially memorized for Part I of the A.P. Exam, whether it is formulas or concepts. This stuff you have to know cold! If you do, you will save a great amount of time in answering questions without having to go through a lot of calculating. Linear Motion Memorize the following formulas: When an object is accelerating there is a SQUARED relation between distance traveled ∆x = v t + 1 at 2 o 2 and the time spent traveling: If the time an accelerating object travels is tripled, the 2 2 v = vo + 2a∆x distance it travels while accelerating will be nine time further than original. It is a DIRECT relation between velocity and time when there is a constant acceleration. v = vo + at The slope of a distance/time graph is the average velocity. The slope of a velocity/time graph is the average acceleration, while the area under the graph is the displacement. SPECIAL CASES of Linear Motion FREE FALL – Substitute the acceleration due to gravity (g) into each of the three linear motion equations. Also remember to use 10 m/s 2 for the value of g in those equations to make quick calculations. Time to fall from rest: rework The first linear motion equation to solve for the time to fall from a certain 2y height and it becomes: t = where y is the height fallen from. g Speed or Velocity as the object strikes the ground from a known height: rework the second linear motion equation to solve for speed as the object strikes the ground and you get: v = 2gh where h is the height the object falls from. PROJECTILE MOTION: You need to remember the two situations: Starting by being thrown horizontally (rolling off a table edge), and being kicked off the ground at an angle (propelled up at an angle.) Horizontal Projection: Remember that the time to fall (from X to Y) is the same as vx 2y X the time of flight . Use: t = . This determines how long g the object is in the air. Y Z Remember that something falling from X to Y takes the same time to move as the object flying from X to Z. To find the distance from Y to Z using: x = vxt where t is the time to fall. Increasing vx Does NOT increase the time of flight, only increasing the height does! And, that is a square root relation. To double the flight time, the height must be increased by four times. If the height is what is doubled, the flight time will only increase by 2 . Y P Regular Projectile Motion: The acceleration is always g and is always downward throughout the flight. X θ Z The velocity at Y is NOT zero, but is the horizontal component of the object’s launch velocity. Determine the time of flight using the vertical components of motion. Hangtime is the total time it’s in the air and it is two times the time it takes to rise to point Y. To find the RANGE of the flight: x = vxt As the angle of launch increases for a given initial launch speed: 1. The horizontal component of the launch velocity decreases. 2. The vertical component of the launch velocity increases. 3. The height the object rises to increases. 4. The time of flight increases. 5. The RANGE (horizontal distance (x)) traveled increases until the launch angle is 45º then it decreases 6. Complimentary angles of launch will give you the same RANGE for the projectile to travel. VECTORS and TRIG Functions 30-60-90 Right Triangle: let’s say two vectors are acting perpendicularly to one another and the angle between one of them and the resultant is either 30º or 60º. They will automatically form a 30-60-90 Rt. Triangle. When that happens there is the following relation between all the sides and parts of the triangle and the vectors. Angle Sine Cosine Tangent 1. Vector B must be 3 times the length of Vector 1 3 1 3 Resultant A. 30 or 2. The Resultant will always be 2 times the length 2 2 3 3 Vector BVector of Vector A. 3 1 3. Memorize the following Trig relations for this 60 3 2 2 60º triangle especially for SINE : Vector A Golden Triangle: 3-4-5 Rt. Triangle: Let’s say that two vectors are acting perpendicularly to each other and they have a relation between their lengths where the ratio of the short to the long leg of the triangle is 3:4 (i.e. 12:16, or 21:28, or 1.5:2). The triangle formed by these two 37º vectors and their resultant make the 3-4-5 Golden Triangle and have the following relation: Resultant = “5” 1. The angle opposite the “3” side will always be Angle Sine Cosine Tangent 37º. 3 4 3 2. The angle opposite the “4” side will always be 37 Vector B Vector = “4” 53º. 5 5 4 53º 4 3 4 3. Memorize the following trig relations for this 53 Vector A = “3” triangle especially for SINE : 5 5 3 Newton’s Laws: 1. Be able to draw the Free-Body Diagram of all the forces acting on an object. 2. USE the Free-Body Diagram to find the Net Force acting on an object. 3. Forces that act Parallel to each other can be added or subtracted to find the net force. Only forces parallel to the direction of motion will create a linear acceleration that is changing the speed. Fnet = ∑ F = mTotal a 4. Forces that act Perpendicular to the motion can change the direction of the motion and may create curved motion. Perpendicular forces create circular motion and centripetal or “radial” acceleration. 5. Friction always acts opposite to the motion or potential motion of the object. 6. The Normal Force is the force of the surface pressing up on the object perpendicular to the surface, not necessarily the ground. 7. Tension is a force! Circular Motion: 2 1. Centripetal Force has a squared relation with the speed the object is moving around the circle mv with. FC = r 2. Centripetal Force has an inverse relation with the object’s radius. v2 3. Centripetal Force is ALWAYS the NET force acting on the rotating object. a = c r 4. Centripetal acceleration is also called radial acceleration. F = ma 5. BOTH centripetal force and centripetal or radial acceleration ALWAYS act toward the center C c of the circle or curve made by the object. 6. The direction of the instantaneous velocity is always perpendicular to the centripetal force and centripetal/radial acceleration. 7. If the speed is NOT constant, then the object is experiencing TANGENTIAL (or linear) acceleration in addition to the radial acceleration. The Total acceleration that the object experiences is the vector sum of these two accelerations. Since they are perpendicular to each other (tangential is ⊥ to centripetal/radial) 2 2 at this means that: aTotal = ac + at and the angle between them is: tanθ = ac 8. Horizontal circle on a rough table top: Set the centripetal force equal to the frictional force (because friction is making it turn!) and the minimum speed for a given centripetal force is v = µgr 9. VERTICAL Circle: Remember that the centripetal force must be F net . A) At top: FC = FA + Fg . At a certain minimum speed F A becomes zero (there is no tension on the rope holding the pail, the passengers in the roller coast “just remain” in their seats, the block “remains in contact” with the track at the top of the loop, etc.) then FA at the top is ZERO and solving for the minimum speed we get: v = gr B) At the Bottom: FC = FA − Fg . F A is the “Tension on the rope, the force the riders in the roller coaster feel, etc.” at the bottom of the circle. Thus: FA = FC + Fg this is showing why you feel heavier at the bottom of the track. Gravitation: The force of gravity is an inverse square relation with the distance between the centers of the objects. When dealing with earth, they will often talk about distances in terms of a number of “Earth Radii” above the earth’s surface. Don’t forget we already stand ONE EARTH RADIUS from the earth’s center. T 2 Planetary Motion: = k . Orbital Period (T) Mean orbital radius (r). This is a constant for any set of r3 objects orbiting another one. Work-Energy-Conservation of Energy-Linear Motion 1. Remember that for work to be done, there must be a change in energy in the system. W = F ⋅ d cosθ Going around a circle “does no work” if it is done at a constant speed. W P = = F ⋅v 2. When a force acts to “do work” if it is not parallel to the motion (displacement) you need t to use the cosine of the angle between the force and displacement vectors (this gives you 1 2 the component of the force acting parallel to the displacement!) KE = 2 mv 3. KE and speed/velocity have a squared relation. 1 2 U Spring = 2 kx A) Remember, if Velocity is changed…square that change to find the change in KE. B) If the KE is changed…take the square ROOT of the change in KE to find the change UGravity = mgh in v.
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