Chapter 2. Kinematic Equations. Problem 1. Kinematic Equations, specific. Motion in One Dimension

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1 Kinematic Equations Chapter Motion in One Dimension The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant You may need to use two of the equations to solve one problem Many times there is more than one way to solve a problem Problem 1 A particle moves along the x axis according to the equation x = t - t, where x is in meters and t is in seconds. (a) What is its velocity at t =.80 s? (b) What is its at t =.80 s? For constant a, v = v + a t xf xi x Can determine an object s velocity at any time t when we know its initial velocity and its Does not give any information about displacement 1

2 For constant, vxi + vxf vx = The average velocity can be expressed as the arithmetic mean of the initial and final velocities For constant, 1 x f = xi + vxit + axt Gives final position in terms of velocity and Doesn t tell you about final velocity For constant, 1 x f = xi + vt = xi + ( vxf + vxi ) t This gives you the position of the particle in terms of time and velocities Doesn t give you the For constant a, v = v + a( x x ) f i f i Gives final velocity in terms of and displacement Does not give any information about the time

3 Problem The driver of a car slams on the brakes when he sees a tree blocking the road. The car slows uniformly with m/s for 4.00 s, making straight skid marks 63.6 m long ending at the tree. With what speed does the car then strike the tree? Graphical Look at Motion velocity time curve The slope gives the The straight line indicates a constant Graphical Look at Motion displacement time curve The slope of the curve is the velocity The curved line indicates the velocity is changing Therefore, there is an Graphical Look at Motion time curve The zero slope indicates a constant 3

4 Problem 3 The position versus time for a certain particle moving along the x axis is shown in the figure below. Find the average velocity in the following time intervals. (a) 0 to s (b) 0 to 4 s (c) s to 4 s (d) 4 s to 7 s (e) 0 to 8 s Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone. It does not depend upon the initial motion of the object Dropped released from rest Thrown downward Thrown upward Problem 4 A position-time graph for a particle moving along the x axis is shown in the figure. The divisions along the horizontal axis represent 1.50 s and the divisions along the vertical axis represent 5.0 m. Determine the instantaneous velocity at t = 6.00 s (where the tangent line touches the curve) by measuring the slope of the tangent line shown in the graph. Acceleration of Freely Falling Object The of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall is g = 9.80 m/s g decreases with increasing altitude g varies with latitude 9.80 m/s is the average at the Earth s surface 4

5 Acceleration of Free Fall, cont. We will neglect air resistance Free fall motion is constantly accelerated motion in one dimension Let upward be positive Use the kinematic equations with a y = g = m/s General Problem Solving Strategy Conceptualize Categorize Analyze Finalize Free Fall Example A ball is thrown vertically upward from the ground with an initial speed of 15.0 m/s. (a) How long does it take the ball to reach its maximum altitude? (b) What is its maximum altitude? (c) Determine the velocity and of the ball at t =.00 s. Answer: (a) 1.53 s; (b) 11.5 m; (c) v = 4.60 m/s, a = 9.80 m/s Problem Solving Conceptualize Think about and understand the situation Make a quick drawing of the situation Gather the numerical information Include algebraic meanings of phrases Focus on the expected result Think about units Think about what a reasonable answer should be 5

6 Problem Solving Categorize Simplify the problem Can you ignore air resistance? Model objects as particles Classify the type of problem Try to identify similar problems you have already solved Problem Solving Finalize Check your result Does it have the correct units? Does it agree with your conceptualized ideas? Look at limiting situations to be sure the results are reasonable Compare the result with those of similar problems Problem Solving Analyze Select the relevant equation(s) to apply Solve for the unknown variable Substitute appropriate numbers Calculate the results Include units Round the result to the appropriate number of significant figures Problem Solving Some Final Ideas When solving complex problems, you may need to identify sub-problems and apply the problem-solving strategy to each sub-part These steps can be a guide for solving problems in this course 6

7 Problem 5 A rock is dropped from rest into a well. The sound of the splash is actually heard.4 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s. 7