Physics 103: Lecture 4 Position & Velocity with constant Acceleration

Physics 103: Lecture 4Position & Velocity with constant Acceleration • Today’s lecture will be on kinematic equations • 1D motion with constant acceleration • free-fall • Introduction to 2D motion Physics 103, Fall 2009, U. Wisconsin

Summary of Concepts (from last lectures) • kinematics: A description of motion • position: coordinates of a point • displacement: x = xf - xichange of position (+ or -) • distance: magnitude of displacement • total distance: sum of all the magnitudes of the displacements (equal to or larger than the displacement) • velocity: rate of change of position (+ or -) • average : x/t = xf - xi/tf - ti • instantaneous: slope of x vs. t • speed: magnitude of velocity, “total” distance/time (equal to or larger than the average velocity) • acceleration: rate of change of velocity (+ or -) • average: v/t = vf – vi/ tf – ti • instantaneous: slope of v vs. t Physics 103, Fall 2009, U. Wisconsin

1D Kinematics Equations for Constant Acceleration Physics 103, Fall 2009, U. Wisconsin

2 More equation for Constant Acceleration Physics 103, Fall 2009, U. Wisconsin

Use of Kinematic Equations - I • Gives position given time, velocity & acceleration • Displacement can be found by calculating x-x0 • Use when you don’t know or need the final velocity • Shows velocity as a function of acceleration and time • Use if don’t know or need the position or displacement • Graphical interpretation  Physics 103, Fall 2009, U. Wisconsin

Lecture 3, Pre-Flight Q.4 Correct x=1/2 at2 Correct v=at An object is dropped from rest. If it falls a distance D in time t then how far will if fall in a time 2t ? 1. D/42. D/23. D4. 2D5. 4D Follow-up question: If the object has speed v at time t then what is the speed at time 2t ? 1. v/42. v/23. v4. 2v5. 4v Physics 103, Fall 2009, U. Wisconsin

Question I I am going to roll the ball down the inclined plane. If the ball reaches mark at distance 1 ft at time t1, when will the ball reach the mark at distance 9 ft? 1. t9 = 9t1 2. t9 = √18 t1 3. t9 = 3t1 Physics 103, Fall 2009, U. Wisconsin

Use of Kinematic Equations - II • Gives position(or displacement) as a function of velocity and time • Use if don’t know or need the acceleration • Gives velocity as a function of acceleration and distance or displacement • Use when you don’t know or need the time Physics 103, Fall 2009, U. Wisconsin

Question II 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. You applied the break hard and stopped right in front of the ducks. What was the acceleration you applied? • xi=0, xf=30 m, vi = 55 mph = (55x1.609) 88.5 km/hr = 24.6 m/s, vf=0, • 0 = vi2 + 2 a xf • a = - vi2/2xf = - 10 m/s2 A large deceleration! How to avoid this sharp change? Physics 103, Fall 2009, U. Wisconsin

Free Fall Principles • Objects moving under the influence of gravity only are in free fall • Objects falling near earth’s surface fall with constant acceleration due to gravity, and indicated by g • g = 9.8 m/s2 • g is always directed downward • toward the center of the earth • Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion • Most common example of constant acceleration Physics 103, Fall 2009, U. Wisconsin

Free-Fall up y x down • constant downward acceleration • g: acceleration due to gravity • same for all bodies: g=9.8 m/s2 • ay = -g = -9.8 m/s2 Summary of Free-Fall Equations y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2gy Physics 103, Fall 2009, U. Wisconsin

Throwing Down Question A ball is thrown downward (not dropped) from the top of a tower. After being released, its downward acceleration will be: 1. greater than g 2. exactly g 3. smaller than g Physics 103, Fall 2009, U. Wisconsin

Lecture 3, Pre-Flight Q. 1 & 2 correct At the top of the path, the velocity of the ball is zero,but the acceleration is not zero. The velocity at the top is changing, and the acceleration is the rate at which velocity changes. Acceleration is the change in velocity. Just because the velocity is zero does not mean that it is not changing. Acceleration is not zero since it is due to gravity and is always a downward-pointing vector. A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: 1. velocity is zero and acceleration is zero2. velocity is not zero and acceleration is zero3. velocity is zero and acceleration is not zero4. velocity is not zero and acceleration is not zero Physics 103, Fall 2009, U. Wisconsin

Free Fall dropping & throwing • Drop • Initial velocity is zero • Acceleration is always ay = -g = -9.80 m/s2 • Throw Down • Initial velocity is negative • Acceleration is always ay = -g = -9.80 m/s2 • Throw Upward • Initial velocity is positive • Instantaneous velocity at max height is 0 • Acceleration is always ay = -g = -9.80 m/s2 vo= 0 (drop) vo< 0 (throw) a = g v = 0 a = g Physics 103, Fall 2009, U. Wisconsin

Correct: v2 = v02 -2gy v0 Dennis Carmen v0 H vC vD Lecture 3, Pre-Flight Q.3 Dennis and Carmen are standing on the edge of a cliff. Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball is moving fastest when it hits the ground? 1. Dennis' ball2. Carmen's ball3. Same On the dotted line: y=0 ==> v2 = v02 v = ±v0 When Dennis’s ball returns to dotted line its v = -v0 Same as Carmen’s v = v0- g t t = 2 v0 / g v = -v0 Physics 103, Fall 2009, U. Wisconsin

Free Fall Scenarios • Is the motion symmetrical? • Then tup = tdown • Then v = -vo • The motion may not be symmetrical  • Break the motion into various segments • Are there symmetrical segments? • Possibilities include • Upward and downward portions • symmetrical portion back to releasepoint and non-symmetrical portion Physics 103, Fall 2009, U. Wisconsin

A battleship simultaneously fires two shells at enemy ships from identical canons. If the shells follow the parabolic trajectories shown, which ship gets hit first? 1. Ship A. 2. Ship B. 3. Both at the same time Higher the shell flies, the longer it takes. What should the captain order if he wants to hit both ships at the same time? A B Physics 103, Fall 2009, U. Wisconsin

Physics 103: Lecture 4 Position & Velocity with constant Acceleration