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Ex. An object moves in ________________ with an ____________________ and an ____________________ . (When graphing v vs. t, the area = _____________ .) vf The extra distance due to acceleration: dB = = = = v vi The distance if there is no acceleration: dA = t t Total distance d: d = d A + dB d = If an object is not accelerating, then ___________ In this case, the last equation becomes: d = d = This last equation is really the same as: v= after it is solved for: since v is ______________ when a = 0, then ______will be the same as _______. One more equation: and these equations we studied before: are all in your _____________ . but: is _______ in PhysRT All of these equations This is called: only describe motion: _____________________ In these equations, all of the quantities are ______________ . This means they have _______________ and _______________ . Ex: If a ball is thrown with an initial speed of 7 m/s, use: vi = ________ if is thrown upward/to right vi = ________ if is thrown downward/to left. Ex. If a rocket moves a distance of 35 m, use: d = ________ if is moves upward/to right d = ________ if is moves downward/to left. Typical physics word problems involve objects moving in ____ direction that: ….begin at one point with an _________ speed or velocity: …. and end at another point with a __________ speed or velocity: ….and move a distance or displacement ____ with an average speed _____ and at an acceleration _____ during a time interval of _____ . Use units to help you solve problems: quantity time distance or displacement speed or velocity acceleration symbol unit abbrev. of unit Word clues: 1. starts at rest ______________ 2. comes to rest _______________ 3. uniform motion __________ , and _____________________ 4. constant velocity _________________________ 5. constant speed in 1 direction ___________ 6. Acceleration will always be ________________ . It may be _________________, _________________ or _____ (still constant, but then _____________). 7. slows down: a and v are in _______________ directions 8. If an object collides with another object (like Earth), then its final speed is _________ ! In this case, "final speed" means the speed ___________________________ it hits. 9. Whatever direction an object is moving is the direction of its _____________________ . 10. If an object changes direction, then, at that instant, only its ___________________________ MUST be zero. Free Fall: •Used to describe the motion of any object that is moving _____________________________ •the only force acting is ________________ •no _____________________ , which is a good approximation if object moves ____________ • motion can be _________________ or in an arc known as a ____________________ • the results are independent of ___________ •All of the equations of __________________can used as long as you use: a = _______ = ___________________= ____________ = _____________on or near Earth’s surface for the time the object is in ________________ . Free fall applies to an object that is… fired up or _________ ___________ down _____ down: from rest: __________: fired _______up: ______________: …only for the time while it is ________________. In all cases: 1. d is _________________if the object ends up __________ the point where it started. 2. d is _________________if the object ends up __________ the point where it started. 3. v is positive if object is going ________________ 4. v is negative if object is going ________________ 5. a is _________________________ I. ______________ motion A. Dropped Objects. Ex 1: A ball is dropped. How far will it fall in 3.5 seconds? given: unknown: equation: Ex. Harry Potter falls freely 99 meters from rest. How much time will he be in the air? given: unknown: equation: Ex. A dinosaur falls off a cliff. What will be its velocity at the instant it hits ground if it falls for 1.3 seconds? equation: given: unknown: A rock that has half the mass of the dinosaur is dropped at the same time. If it falls for the same time, what will its final speed be? Which will hit the ground first? B. Objects Fired Up or Down. Ex. A ball is tossed up with an initial speed of 24 meters per second. How high up will it go? given: equation: unknown: What total distance will it travel before it lands? What will be its resultant displacement when it lands? For a ball fired or thrown straight up: 1._______ d each second on way up 2.______ d each second on way down 3. tup = _____________ 4. ttotal = _______ = __________ 5. vtop =__________ 6. atop= __________ 7. speedup = _______________ 8.If object falls back to its original height, then: vf =______ Ex. Mr. Butchko is fired directly up with an initial speed of 55 meters per second. How long will he be in the air? given: equation: unknown: How much time did he spend going up? Ex. A shot put is thrown straight down from a cliff with an initial speed of 15 m/s. How far must it fall before it reaches a speed of 35 m/s? given: unknown: equation: B. Graphical analysis: use a ≈ _____________ Ex: ball dropped from rest v (m/s) t (s) 1 -10 -20 2 3 t (s) 0 1 2 3 -30 4 -40 d v a (m) (m/s) (m/s2) time 0s total d 0m 1s 2s 3s See any patterns? velocity Ball dropped: vectors vs. scalars displacement distance d d t velocity speed v t v t t acceleration acceleration a t a t Ex: ball thrown straight up with vi = 30 m/s t (s) 0 1 2 3 4 5 6 d v a (m) (m/s) (m/s2) 0 30 v (m/s) 30 20 10 1 2 3 4 5 6 -10 -20 -30 slope = ______________ throughout t (s) Coming down: Going up: v time 3s 3s 2s 4s 1s 5s 6s 0s time v At what time is the ball at its highest point? t= What are the v and a at that time? v= a= How do the the last 3 sec of this example compare to the example of a ball dropped from rest? What will the graph of speed vs. time look like? 30 20 10 1 2 3 4 5 6 t (s) II. Understanding Velocity in _________________ . When an object is moving ____________________ as well as _____________________ , its velocity has ______ and ______ components ( __________). In this section, you will study A/ a new way to ______ vectors, and B/ how a velocity vector can be _________________ (broken up into parts). A. Adding Vectors. B Old way: _______________method: A New way: _____________________ method: •draw the 2 vectors as if they come from a ___________________ (see below). •draw a _____________________ using the 2 vectors as sides •The resultant R is the _____________________ drawn from the point A point: B Note: R is _____________ ____________ in the old and new methods Ex: Add and Head to tail: Ex: Add Head to tail: Parallelogram: and Parallelogram: Ex: A train is moving at 50. m/s west. A cannon on the train is fired straight up with an initial speed of 40. m/s. Determine the resultant velocity with respect to someone on the ground. Use the ____________________ method. mag: speed = 40 m/s dir: tanq = 50 m/s q= B. Resolving vectors: Any vector can be _________________ (broken down) into ______________________ (parts) Steps (after drawing the vector itself): 1. Draw ________________ from the tail end of the vector. This is often done for you. 1. Draw __________________from the head of the vector that are _______________to each of the axes 3. Draw the ___________________vectors along the axes, starting at the axes _______________ and ending at the ______________________. v Ex: The components vx and vy are also _____________ . If they are added back together, you will get the___________________ . To determine components, you can either: 1/ set up a scale and _______________directly, or 2/ use ____________ functions. Ex: Using a scale. What is the scale used in the diagram at right? 1 cm = ____ m/s Measure vx and vy: vx = _____ cm = ______ m/s vy = _____ cm = ______ m/s v= m/s Ex: Using trig functions. v In the Math section of your PhysRT: A can be_________________, not just velocity v. Notice also: speed v = ____________ (Pythag. Thm.) tan q = __________ q = tan-1 (__________) Ex: A ball is launched into the air with an initial speed of 46 m/s at an angle of 300 to the horizontal. Find the x- and y- components of the initial velocity. vy = vsinq = = = vx = vcosq = = = Note: 1. Vectors vx + vy = ____ b/c ____________________________ 2. The magnitudes 40 + 23 _____________________________ 3. (vx2 + vy2)1/2 = (402 + 232)1/2 = ______ = ________________ C. Resolution is the __________________of Addition: Addition: Resolution: v Head to tail: Parallelogram: gives you components: Ex. The same vector can be resolved into________________ components, depending on how the __________ are chosen. This v: v has these ______________ Use the same v, but now _________ the axes: …that add up to v: …still add up to the ________ v v …but these new _____________ And the axes need not be ___________________ : ________ v …again add up to the ________ v. …with new _____________ Any vector can be resolved into an ________________ number of component pairs. III. _______________________Fired Projectile: A projectile (object) is launched horizontally with an __________________ from a height ______ . Assume no_____________________. The time in the air before landing is called the ____________________. horizontal distance traveled = _____________ Ex. Ball 1 is________________. Ball 2 is fired __________________ 1 2 vi Both reach the ground ______________________ , regardless of 2's __________________ or ________. The y motion is ______________ of the x motion. Remember: The time it takes a ______________ fired projectile to fall is ______________ the time it takes a ______________ ball to fall from the same height. With no ____________________, only the force of ___________ acts on the object: vi The trajectory (path) is a________________. Air resistance acts in the direction _____________ to its velocity. This __________________ its range. vi The trajectory is _____ ___________ a parabola. 1.Since the object moves in 2 dimensions, each d, v and a must be replaced by their components: For x motion: d, v, a _______________ For y motion: d, v, a _______________ 2. No air resistance only ____________ , which is _______________ , in the ____ direction. There is no _________________ force. Because of this, the only acceleration a is purely vertical: ay = ___________ ax =____________ 3. The initial velocity is purely _________________: viy = ___________ (initially, no y _______________) vi = ________ ("horizontally fired") Horizontal (x) motion: ax =______ displacement: d = v t + ½ at2 i dx = dx dx = dx = velocity: vf = vi +at vfx = vfx = t vfx vfx = The x motion is _________________. t Vertical (y) motion: viy = ____ & ay = __________ displace2 d = v t + ½ at i ment: dy = dy = |dy| dy = velocity: vf = vi +at vfy = vfy = vfy = t |vfy| t The y motion is same as for a _________________. Summary: Horizontal (x) motion: Vertical (y) motion: •____________ motion •______________motion •____________ x-speed • same as for a ball ____________________ dx = dy = vfx = vfy = ax = ay = Ex. A 68-kg clown is fired from horizontal cannon with an initial speed of 40. m/s from a height of 25 m. What is her time of flight? 40. m/s 25 m Given: m= vix = viy = ay = ax = dy = Unknown: Equation: What is her range? What is the x-component of her velocity after 1.5 s? What is her acceleration after 1.5 s? Recalculate the new time of flight and range if she is fired with an initial speed of ____________ Time of flight: range: Ex. A dart fired horizontally strikes a target a distance of 0.15 m below where it is aimed. 0.15 m blow gun What was its time of flight? viy = ay = Given: ax = Equation: dy = Unknown: If the target was 9.0 m away from the gun, what was its initial speed? horizontal motion -___________ vertical motion – ______________ a= combined motion -________________ vi 1s 2s 3s 4s 1s 2s 3s 4s Look at how the velocity changes: vi 1s 2s 3s 4s 1s 2s 3s 4s The x-component of v is ________________ The y-component of v __________________ Resultant velocity magnitude (speed) ______________ ___________________to trajectory IV.Projectile fired __________________________ with an initial _________________ Assume no _________________. The only force acting on the projectile is _________ . This means the acceleration is ____________, ______________ vtop ______, atop = ___________ The velocity is always __________ the path To solve the problem, vi must be ____________ into its horizontal (vix) vi viy = _______ and vertical (viy) _____________________. q vix =__________ Where: vi = _______________ is the initial speed, and q = __________________ is the angle. There are _____ simultaneous motions: For ___ motion, use: _____________________ For ___ motion, use: _____________________ A. The horizontal motion is determined by ___ = _______ . Because there is _______ horizontal force, vix __________________ _____________ x-motion. acceleration: ax = ax t velocity: vfx = vfx t displacement: dx = dx t B. Vertical motion is determined by ___ = _______ . Because of ____________, the y motion is like a ball thrown _______________ with an initial speed ____ . acceleration: ay = = velocity: ay t vfy = = vfy t displacement: dy = = dy t Ex 1: Ms. Rudd is fired out of a cannon at a speed of 75 m/s and at an angle of 370 to the horizontal. 75 m/s 370 vix = vicosq = viy = visinq = To determine how high up she goes and how long she is in the air, "pretend" she is fired __________ ___ but with an initial speed = _____ = __________ Given: viy = 1st Unknown: ay = vfy = 2nd Unknown: How far up? How long is she in the air? Because we chose vfy = ___ , this t represents the time to _________________ . To get the total time of flight, we must _____________________ . So, the total time t = _______ s. You could get this time directly if you assume vfy = __________ . Then: To determine her range, you must assume her x motion is ____________ at vi = ____ = _______ . Given: vix = Unknown: ax = t= Notice that the ___________ time is used here! With no ____________________, only the force of ___________ acts on the object: vi The trajectory (path) is a________________. Air resistance acts in the direction _____________ to its velocity. This _____________ its max. height ' and range. The trajectory is _______ ________________________ vi Ex 2: A graphical example On way up: horizontal motion -________________ vertical motion –ball thrown________________ combined motion -______________ 3s 2s ay = 1s vi 1s 2s 3s coming down: The motion is exactly the same as that of a projectile which is _______________________ : 3s 4s 5s 6s 3s 4s 5s 6s Velocity vectors: going up 3s 2s 1s vi viy vix 1s 2s 3s resultant velocity found by adding ____ and ____ is _______________ to the parabola is = ________ (NOT = ____ ) at the max. height. Velocities coming down: 3s v 4s 5s 6s 3s 4s 5s 6s Notice the ______________ with going up The effect of changing ___ on the trajectory. Assume all are fired with ________________ vi. Which q results in longest range? Which results in highest trajectory? In longest time in air? Which is a parabola? As q increases, the ___ component of vi increases. Because of this: total time in air ________________ , and maximum height ______________ angle ________________________ angles have the same range. compl. angle angle with greater…. range time of flight max. height 80 60 47 Range as a function of q, assuming range for 450 is 100. Fill in the rest: 100 75 50 25 0 15 30 45 60 75 90 q