stopping distance : Physics 201, Spring 2011 Lect. 4 Example 2 13, The Flying Cap (p 44) Example 2 13, The Flying Cap (p 44) 1/19/11

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1 Physics 201, Spring 2011 Lect. 4 Chapter 2: Mo;on in One Dimension (examples and applica;ons) stopping distance : You were driving on a country road at an instantaneous velocity of 55 mph, east. You suddenly saw a family of ducks walking across the road 30 m ahead of you (the stopping distance). You applied the break hard and stopped right in front of the ducks. What was the accelera;on you applied? x i =0, x f =30 m, v i = 55 mph = (55x1.609) 88.5 km/hr = 24.6 m/s, v f =0, 0 = v i a x f a = v i2 /2x f = 10 m/s 2 A large decelera;on! How to avoid this sharp change? Example 2 13, The Flying Cap (p 44) A cap is tossed in the air. It starts at height y 0 with a velocity v 0y =14.7 m/s upward. How long does it take for the cap to reach its highest point? Example 2 13, The Flying Cap (p 44) A cap is tossed in the air. It starts at height y 0 with a velocity v 0y =14.7 m/s upward. How long does it take for the cap to reach its highest point? Highest point is when the velocity is zero. v y (t) = v 0y + a y t f t f = (v y (t)-v 0y )/a y = ( m/s)/(-9.81 m/s 2 ) = 1.50 s How high does the cap go? How high does the cap go? The cap goes distance: y f = v 0y t f + (1/2)a y t 2 f = (14.7 m/s)(1.50 s) + (1/2)(-9.81 m/s 2 )(1.50 s) 2 = = 11.0 m by W.H. Freeman and Company 1

2 Example 2 13; Height and velocity of the flying cap Height function of t quadratic. Gebng velocity from accelera;on by integra;on Velocity function linear. Acceleration = const. Constant acceleration: v(t) = v 0 + a (t t 0 ) (The only tricky part is putting in the limits of integration.) Gebng posi;on from velocity by integra;on The displacement is the integral of the velocity (the limit of the sum of areas of the rectangles and it can be negative!). Constant acceleration: x(t) = x 0 + v 0 (t t 0 ) + (1/2) a (t t 0 ) 2 (The only tricky part is putting in the limits of integration.) Phys 1/19/11 201, Spring

For constant accelera;on, as we saw before: Problem 2.31 Reginald and Josie go the same distance within the same ;me. What is rela;onship between v J and v max?

3 For constant accelera;on, as we saw before: Problem 2.31 Reginald and Josie go the same distance within the same ;me. What is rela;onship between v J and v max? Area = x(t) = v 0x t + (1/2) a t 2 v J Phys 1/19/11 201, Spring 2011 Phys 1/19/11 201, Spring 2011 Problem 2.31 Reginald and Josie go the same distance within the same ;me. What is rela;onship between v J and v max? Problem write algebraic expressions for x(t), v x (t), and a x (t) v J Distance traveled is area under v-versus-t curve. For Josie, this area is v J t f. For Reginald, this area is t f v max / 2. The areas are equal when v max =2v J /19/11 by W.H. Freeman and Phys 201, Spring 2011 Phys 1/19/11 201, Spring 2011 Company 3

4 Problem write algebraic expressions for x(t), v x (t), and a x (t) Mo;on with ;me varying accelera;on An object undergoes a linearly increasing acceleration a(t) = 10 t m/s 2 along a straight line. It starts at the origin x(0)=0 with a velocity of 2 m/s. Where is its position at time t=4 seconds? Reading from graph, v x (t) = v 0 + At, with v 0 =50 m/s and A=-10 m/s 2. Acceleration is derivative of velocity: a x (t)=a=-10m/s 2. Position is integral of velocity: x(t) = x(0)+v 0 t + ½At 2 = 50t - 5t by W.H. Freeman and Company Mo;on with ;me varying accelera;on An object undergoes a linearly increasing acceleration a(t) = 10 t m/s 2 along a straight line. It starts at the origin x(0)=0 with a velocity of 2 m/s. Where is its position at time t=4 seconds? In this problem the acceleration is NOT constant. So must do the integrals explicitly. Which graph of v versus t best describes the mo;on of a par;cle with posi;ve velocity and nega;ve accelera;on? 4

5 Which graph of v versus t best describes the mo;on of a par;cle with posi;ve velocity and nega;ve accelera;on? A car accelerates uniformly from a velocity of 10 km/h to 30 km/h in one minute. Which graph best describes the motion of the car? A car accelerates uniformly from a velocity of 10 km/h to 30 km/h in one minute. Which graph best describes the motion of the car? The accelera;on of a vehicle is given by a(t) = At where A is a constant. Its velocity as a func;on of ;me is (v o is a constant) 5

6 The accelera;on of a vehicle is given by a(t) = At where A is a constant. Its velocity as a func;on of ;me is (v o is a constant) Conceptual ques;on: velocity and accelera;on A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: a. velocity is zero and acceleration is zero b. velocity is not zero and acceleration is zero c. velocity is zero and acceleration is not zero Conceptual ques;on: velocity and accelera;on A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: a. velocity is zero and acceleration is zero b. velocity is not zero and acceleration is zero c. velocity is zero and acceleration is not zero correct At the top of the path, the velocity of the ball is zero, so that it doesn t go any higher. The velocity at the top is s;ll changing, and the accelera;on is the change in velocity. Just because the velocity is zero does not mean that it is not changing. So the accelera;on is NOT zero. Accelera;on is due to gravity, and is constant, and always a downward poin;ng vector. 6

Phys 201 Fall 2009 Thursday, September 10, 2009 & Tuesday, September 15, Chapter 2: Mo>on in One Dimension

Phys 201 Fall 2009 Thursday, September 10, 2009 & Tuesday, September 15, 2009 Chapter 2: Mo>on in One Dimension The first exam will be 5:45pm 7pm on Tuesday, September 29 Please let us know as soon as