How quickly can you stop a car? You are driving along a road at a constant speed V 0. You see a stop sign, you step on the breaks, and the car slows down with constant decelera;on a. 1. How much ;me does it take to stop? 2. How far do you travel before you come to a stop? x 0 x 1 When you hit the brakes Where you stop 1
Constant accelera;on Velocity changes at at constant rate: a v = v + at 0 is the slope of v(t): a = v ( t ) v 0 t 0 v( t) v 0 = at 0 t The average velocity is The distance traveled is ( ) + v 0 v = v t = v 0 + at 2 2 x = x 0 + v t 0 ( ) x = x 0 + v 0 t + 1 2 at 2 2
Free Fall Free fall is a good example for a one dimensional problem Gravity Things accelerate towards earth with a constant accelera;on g = 9.8m/s 2 towards the earth We ll use Gravity a lot! 3
Throw a Ball up You throw a ball upward into the air with ini;al velocity V 0. Calculate: a) The ;me it takes to reach its highest point (the top). b) Distance from your hand to the top c) Time to go from your hand and come back to your hand d) Velocity when it reaches your hand e) Time from leaving your hand to reach some random height h. 4
Speeder A speeder passes a police officer siyng by the side of the road and maintains her constant velocity V. The officer immediately starts to chase the speeder. He starts from rest with constant accelera;on a. How much ;me does it take to catch the speeder? How far does he travel to catch the speeder? What is his final speed? Police Officer Speeder X 5
Even the signs of posi;on and ;me may not be related: consider a pendulum: posi;on x(t) velocity accelera;on
Jules throws a baseball straight up, with an ini;al velocity of 10 m/s. a) How high does it go before coming momentarily to a stop? b) How much ;me passes before it falls and hits him on the head? 1. Draw a picture of the situa=on, and your visualiza=on of the trajectory of the ball. 2. Model the problem using the tools of physics. Choose either to analyze it by wri=ng equa=ons of mo=on, or by using average velocity. 3. Solve the model to obtain the quan==es asked in the problem.
2. Two soccer players, Bob and Jane, are 30 m apart. At the same instant they start to run towards each other. Bob runs with a constant accelera;on of 0.60 m/ s 2 and Jane runs with a constant speed of 2 m/s. a) How far does Jane run un;l they collide? b) What is Bob's speed just before they collide? 1. Draw a picture that shows what is happening in the problem. Give names to the kinema=c quan==es in the problem. For vectors, show their direc=on. 2. Write equa=ons of mo=on for Bob and Jane. Define the moment when they collide. 3. Solve for the quan==es asked in the problem.
Posi;on, Velocity, Accelera;on are measured in a reference frame! r!! R y!! r z! r!! r " = R" x
The same applies when an object is observed in to frames that are co- moving! v!! v! v!! v " = V"!! V
Mo;on in Two Dimensions We will use the same tools of kinema;cs to describe mo;on in two dimensions. A common kind of problem has constant accelera;on in one dimension, uniform mo;on in the other dimension: Example: Basketball shot parabolic trajectory The key to this kind of problem is to analyze the mo;on in each coordinate direc;on separately, then put the two descrip;ons together to describe the mo;on in 2- D. 11
3. A passenger is looking out the window of a train as it travels E at a velocity of 20 m/s on a rainy day. The raindrops are falling ver;cally down, but as she sees them they appear to make an angle of 30 o W of ver;cal. What is the speed of the raindrops? a) as seen from the train? b) as seen by an observer standing on the ground outside? 1. Draw a picture that shows what is happening in the problem. Give names to the kinema=c quan==es in the problem. For vectors, show their direc=on. 2. Write vector equa=ons rela=ng the veloci=es in the problem. 3. Solve for the quan==es asked in the problem.
Ball Dropping, Ball Tossed Horizontally Analyze ver;cal and horizontal mo;ons separately!!! A y = g (downwards) A x = 0 V x = Constant for both cases uniform mo;on! V x = 0 V x >0 13
Superposi=on: Describe the x, y mo;ons separately Prove that an object projected horizontally will reach the ground at the same ;me as an object dropped ver;cally. Study the spreadsheet of the 2- D trajectory of an ar;llery shell on e- learning. Experiment with it to determine the angle that gives maximum range. 14
The case of the dog who dived from the cliff A ver;cal cliff is located at the edge of a lake, with its ledge 5 m above the water. A dog runs horizontally off the cliff. He lands 6 m from the shore. What was his speed as he leapt from the cliff? How much ;me was he in the air? There is a method for analyzing problems in physics. We are going to apply that method to this problem: 1. Visualize the problem draw a picture, label the features that are at issue in the problem. 2. State what you know and what you are asked to find. 3. Write a model of the problem in the language of physics. 4. Solve for what you are asked to find in the problem.
1. Visualize the problem v 0 h = 5m d = 6m 2. State what you know, what you must find. The dog moves in 2 dimensions: x and z. There is no accelera;on in x: uniform mo;on. There is constant downward accelera;on g.
3. Model the problem: Set up equa;ons of mo;on in x, z General equa;on of mo;on with constant accelera;on: x = x 0 + v 0 t + 1 2 at 2 Horizontal mo;on: x 0 = 0, v x0 =?, a x = 0 x = v 0 t Ver;cal mo;on: z 0 = 5m, v z0 = 0, a z = - 9.8m/s 2 z = (5m)+ 1 ( 9.8m / s2)t 2 2 The dog hits the water when (x,z) = (6m,0)
4. Solve the problem There are two unknowns in the equa;ons of mo;on: v x0 and T (the ;me when the dog hits the water). Two equaoons in two unknowns you can solve for both unknowns. x: 6m = v 0 T z: T = 6m v 0 v 0 = 6m ( ) 0 = 5m ( ) + 1 2 (9.8m / s 2 ) 2 5m ( ) ( 9.8m / s2)t 2 0 = (5m)+ 1 2! ( 9.8m / ) 6m $ s2 # & " % v 0 = 6m / s T = 6m 6m / s =1s 2