OneDimensional Motion Problem Solving Identify Read problem carefully

One-Dimensional Motion

Problem Solving Identify – Read problem carefully – Make a sketch – Strategize Set up – Choose appropriate equations Execute – Solve the equations Evaluate

Scalars and Vectors Scalar – a number, (usually) with units – Mass, length, time Vector – a quantity with a numerical value (with units) and direction

Coordinate System

Distance and Displacement

Concep. Test 2. 1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? 1) yes 2) no

Concep. Test 2. 1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same 1) yes 2) no displacement? Yes, you have the same displacement. Since you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement. Follow-up: Have you and your dog traveled the same distance?

Average Speed and Velocity Average speed = distance / elapsed time Average velocity = displacement / elapsed time

Concep. Test 2. 3 Position and Speed 1) yes If the position of a car is zero, does its speed have to be zero? 2) no 3) it depends on the position

Concep. Test 2. 3 Position and Speed 1) yes If the position of a car is zero, does its speed have to be zero? 2) no 3) it depends on the position No, the speed does not depend on position; it depends on the change of position. Since we know that the displacement does not depend on the origin of the coordinate system, an object can easily start at x = – 3 and be moving by the time it gets to x = 0.

Concep. Test 2. 6 b Cruising Along II You drive 4 miles at 30 mi/hr and 1) more than 40 mi/hr then another 4 miles at 50 mi/hr. 2) equal to 40 mi/hr What is your average speed for 3) less than 40 mi/hr the whole 8 -mile trip?

Concep. Test 2. 6 b Cruising Along II You drive 4 miles at 30 mi/hr and 1) more than 40 mi/hr then another 4 miles at 50 mi/hr. 2) equal to 40 mi/hr What is your average speed for 3) less than 40 mi/hr the whole 8 -mile trip? It is not 40 mi/hr! Remember that the average speed is distance/time. Since it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr. Follow-up: How much further would you have to drive at 50 mi/hr in order to get back your average speed of 40 mi/hr?

x vs. t graph

Average velocity on x vs. t graph

Instantaneous Velocity

Concep. Test 2. 13 a The graph of position versus Graphing Velocity I 1) it speeds up all the time 2) it slows down all the time for a car is given below. 3) it moves at constant velocity What can you say about the 4) sometimes it speeds up and velocity of the car over time? sometimes it slows down 5) not really sure x t

Concep. Test 2. 13 a The graph of position versus Graphing Velocity I 1) it speeds up all the time 2) it slows down all the time for a car is given below. 3) it moves at constant velocity What can you say about the 4) sometimes it speeds up and velocity of the car over time? sometimes it slows down 5) not really sure x The car moves at a constant velocity because the x vs. t plot shows a straight line. The slope of a straight line is constant. Remember that the slope of x versus t is the velocity! t

Concep. Test 2. 13 b Graphing Velocity II 1) it speeds up all the time The graph of position vs. 2) it slows down all the time for a car is given below. 3) it moves at constant velocity What can you say about the 4) sometimes it speeds up and velocity of the car over time? sometimes it slows down 5) not really sure x t

Concep. Test 2. 13 b Graphing Velocity II 1) it speeds up all the time The graph of position vs. 2) it slows down all the time for a car is given below. 3) it moves at constant velocity What can you say about the 4) sometimes it speeds up and velocity of the car over time? sometimes it slows down 5) not really sure The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity! At large t, the value of the position x does not change, indicating that the car must be at rest. x t

Acceleration

v vs. t graphs

Graphical Interpretation of Acceleration

Acceleration and Deceleration

Demo Prediction You’ve seen the cart move up and down the table. At the highest point in the cart’s motion: 1) both v = 0 and a =0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure

Concep. Test 2. 14 a v versus t graphs I 1) decreases Consider the line labeled B in 2) increases the v versus t plot. How does 3) stays constant the speed change with time 4) increases, then decreases for line B? 5) decreases, then increases v A t B

Motion with constant acceleration

Constant Acceleration Equations of Motion

TAPPS Think Aloud Pair Problem Solving Get into pairs – choose one person as THINKER and one as LISTENER Say everything that comes to mind as you’re trying to solve the problem LISTENERS should only intervene when the THINKER is making a mistake

Example Two cars drive on a straight highway. At time t = 0, car 1 passes mile marker 0 traveling due east with a speed of 20. 0 m/s. At the same time, car 2 is 1. 0 km east of mile marker 0 traveling at 30. 0 m/s due west. Car 1 is speeding up with an acceleration of magnitude 2. 5 m/s 2 and car 2 is slowing down with an acceleration of magnitude 3. 2 m/s 2 – Write x vs. t equations of motion for both cars – At what time do the cars pass next to one another?

TAPPS Example When a chameleon captures an insect, its tongue can extend 16 cm in 0. 1 s. – Find the magnitude of the tongue’s acceleration, assuming it is constant.

Free Fall

Air Resistance

Lava Bombs

TAPPS Problem A glaucous-winged gull, ascending straight upward at 5. 20 m/s, drops a shell when it is 12. 5 m above the ground. – What is the magnitude and direction of the shell’s acceleration just after it is released? – Find the maximum height above the ground reached by the shell – How long does it take for the shell to hit the ground? – What is the speed of the shell at this time?