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Motion Along a Straight Line Physics chapter 2 Physics chapter 2 1 Definitions Mechanics – the study of the relationships among force, matter and motion Kinematics – the description of motion Dynamics – the relation of motion to its causes Physics chapter 2 2 Average velocity Vector quantity Change in x (displacement) divided by change in time x2 x1 x vav t2 t1 t means “change in” Always take final minus initial Physics chapter 2 3 Average velocity Only depends on total displacement Not the details of the motion On a x-t graph (position as a function of time), it is the slope of the line connecting the starting point and the ending point. Physics chapter 2 4 Sign of velocity The sign (positive or negative) of velocity indicates direction It depends on how you set up your coordinate system You can choose your coordinate system, but you must be consistent. Since t is always positive, v has the same sign as x Physics chapter 2 5 Velocity units Must be displacement (length) units divided by time units m/s km/h mi/h ft/s Physics chapter 2 6 Example Jeff leaves his house at 10 am and travels 240 km east. He stops for lunch at noon. At 12:30 he gets back in his car and travels another 100 km east. He arrives at his destination at 1:30 pm. What was his average velocity for the first part of his trip, the second part of his trip, and the whole trip? Physics chapter 2 7 Example First part of trip t1 10 : 00 t2 12 : 00 240 km x vav 2.0 h t x 240 km km 120 h Physics chapter 2 8 Example Second part of trip t1 12 : 30 vav. x t2 1 : 30 x 100 km 1h t x 100 km km 100 h Physics chapter 2 9 Example Whole trip t1 10 : 00 vav , x t2 1 : 30 x 340 km 3.5 h t x 340 km km 97 h Physics chapter 2 10 Instantaneous velocity Can tell us the details of a particle’s motion at a given time. In physics, an instant has no duration – a single point in time Physics chapter 2 11 Instantaneous velocity Take the limit of average velocity as t goes to zero The derivative of x with respect to t x dx v lim t 0 t dt Physics chapter 2 12 Instantaneous velocity On an x-t graph, it is the slope of the tangent line to the curve at a given point In other words, the derivative Physics chapter 2 13 vocab clarification “velocity” refers to instantaneous velocity unless stated otherwise “speed” is distance divided by time – a scalar – has nothing to do with direction Physics chapter 2 14 Example A car travels along a straight road so that its distance from the starting point is given by m 2 m 3 xt 1.50 2 t 0.050 3 t s s What is its instantaneous velocity at t = 3 s? Physics chapter 2 15 Example m m 2 3 x3 s 1.50 2 3 s 0.050 3 3 s s s x3 s 12.15 m t 0.1 s x3.1 s 12.92545m 12.92545 - 12.15 v 0.1 v 7.75 m s t 0.01 s x3.01s 12.226605m 12.226605 - 12.15 v 0.01 v 7.66 m s Physics chapter 2 16 Example t 0.001 s x3.001s 12.15765m 12.15765 - 12.15 v 0.001 v 7.65 m s t 0.0001 s x3.0001s 12.150765m 12.150765 - 12.15 v 0.0001 v 7.65 m s Physics chapter 2 17 Shortcut with calculus m 2 m 3 xt 1.50 2 t 0.050 3 t s s m m 2 dx 21.50 2 t 3 0.050 3 t v s s dt m m 2 21.50 2 3 s 3 0.050 3 3 s s s m 7.65 s Physics chapter 2 18 Average acceleration A vector quantity The rate of change of velocity with time v2 v1 v aav t2 t1 t Physics chapter 2 19 Instantaneous acceleration The derivative of v with respect to t v dv a lim t 0 t dt Physics chapter 2 20 Acceleration on a v-t graph Average acceleration is the slope of a line connecting a beginning and ending point. Instantaneous acceleration is the slope of a tangent line at a given point. Physics chapter 2 21 Acceleration on a x-t graph If the graph is concave up, If the graph is concave down, If the graph has no curvature (an inflection point), • Then the acceleration is positive. • Then the acceleration is negative. • Then the acceleration is zero. Physics chapter 2 22 Sign of acceleration The sign of the acceleration doesn’t tell us whether the object is speeding up or slowing down. We also need to know the sign of the velocity Physics chapter 2 23 Sign of acceleration If v and a have the same sign, Then the object is speeding up. If v and a have opposite signs, Then the object is slowing down. Physics chapter 2 24 Position and acceleration Acceleration is the second derivative of position dv a dt dx v dt d dx d x a 2 dt dt dt 2 Physics chapter 2 25 Very important! Motion with constant acceleration v v0 at 1 2 x x0 v0t at 2 v v 2ax x0 2 2 0 v0 v x x0 t 2 Physics chapter 2 26 CAUTION These equations only work for constant acceleration along a straight line! Physics chapter 2 27 Example On a highway at night you see a stalled vehicle and brake your car to a stop with an acceleration of magnitude 5 m/s2. If your initial speed is 30 m/s (about 67 mph), what is your stopping distance? Physics chapter 2 28 Example m a 5 2 s m v0 30 s m v0 s x0 0 m v v 2ax x0 2 2 0 0 v 2ax 2 0 Physics chapter 2 29 Example 2ax v 2 0 2 0 v x 2a 2 m 30 s x m 2 5 2 s Physics chapter 2 30 Example 2 m 900 2 s x m 10 2 s x 90 m This is about 295 feet. Physics chapter 2 31 Example A motorist passes a school-crossing corner, where the speed limit is 10 m/s, traveling with constant velocity of 20 m/s. A police officer on a motorcycle, stopped at the corner, starts off in pursuit with constant acceleration of 2.0 m/s. How much time elapses before the motorcycle overtakes the car? How fast is the motorcycle traveling at this time? Physics chapter 2 32 Free fall Galileo proposed that all objects fall with the same acceleration, regardless of their weight. Physics chapter 2 33 Acceleration due to gravity When the effects of air can be neglected, Galileo is right. The magnitude of the acceleration due to gravity is denoted with the letter g. Near the surface of the earth, g = 9.8 m/s2. Physics chapter 2 34 Example While standing in an elevator, you see a screw fall from the ceiling. The ceiling is 3 m above the floor. The elevator is not moving. How long does it take for the screw to hit the floor? Physics chapter 2 35 Example 1 2 x x0 v0t at 2 x0 x0 v0 0 a g 1 2 0 x0 gt 2 Physics chapter 2 36 Example 1 2 gt x0 2 2 x0 t g 23 m t m 9.8 2 s t 0.612s 2 2 2 2 t 0.78 s Physics chapter 2 37 Example What if the elevator is moving upward with a constant acceleration of magnitude 4.0 m/s2? Physics chapter 2 38 Example 1 2 x f x0 f v0 f t a f t 2 1 2 xs x0 s v0 s t as t 2 x0 f 0 xs as g v0s v0 f v0 xf x0 Physics chapter 2 39 Example 1 2 x f v0t a f t 2 1 2 xs x0 s v0t gt 2 at timet x f xs 1 1 2 2 v0t a f t x0 s v0t gt 2 2 Physics chapter 2 40 Example 1 1 2 2 a f t x0 s gt 2 2 1 1 2 2 a f t gt x0 s 2 2 1 a f g t 2 x0 s 2 Physics chapter 2 41 Example 2 x0 s t af g 2 1 m a f 4.0 2 s m g 9 .8 2 s 23.0 m 2 t1 m m 4.0 2 9.8 2 s s Physics chapter 2 x0s 3.0 m 42 Example 6.0 m t m 13.8 2 s 2 t 0.435s 2 2 t 0.66 s Does it make sense that the time is shorter? Why or why not? Physics chapter 2 43 Relative velocity No homework problems, just a short discussion. When you are moving, the velocity of something looks different than when you are standing still. Physics chapter 2 44