Kinematics in One Dimension

Transcription

1 Kinematics in One imension Return to Table of ontents Kin in One imension Review of 1 Kinematics Kinematics is the description of how objects move with respect to a defined reference frame. isplacement is the change in position of an object. verage speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time. Instantaneous velocity is the limit as the time becomes infinitesimally short. verage acceleration is the change in velocity divided by the time. Jul 23 1:46 PM 1

2 Review of 1 Kinematics Instantaneous acceleration is the limit as the time interval becomes infinitesimally small. There are four equations of motion for constant acceleration, each requires a different set of quantities. v = v o + at x = x o + v o t + ½at 2 v 2 = v o 2 + 2a(x x o ) v = v + v o 2 Jul 23 1:46 PM 1 Starting from rest, you accelerate at 4.0 m/s 2 for 6.0s. What is your final velocity? v = vo + at v = 0 + 4(6) v = 24m/s ug 14 2:24 PM 2

3 2 You have an initial velocity of 3.0 m/s. You then experience an acceleration of 2.5 m/s 2 for 9.0s; what is your final velocity? v = v o + at v = (9) v = 19.5m/s ug 14 2:24 PM 3 How much time does it take to come to rest if your initial velocity is 5.0 m/s and your acceleration is 2.0 m/s 2? v = vo + at 0 = 5 + 2t t = 2.5s ug 14 2:24 PM 3

4 4 n object moves at a constant speed of 6 m/s. This means that the object: Increases its speed by 6 m/s every second ecreases its speed by 6 m/s every second oesn t move Has a positive acceleration Moves 6 meters every second Jul 19 1:46 PM 5 snapshot of three racing cars is shown on the diagram. ll three cars start the race at the same time, at the same place and move along a straight track. s they approach the finish line, which car has the lowest average speed? ar I ar II ar III ll three cars have the same average speed More information is required Jul 19 1:46 PM 4

5 Motion at onstant cceleration In physics there is another approach in addition to algebraic which is called graphical analysis. The following formula v = v graph. 0 + at can be interpreted by the We just need to recall our memory from math classes where we already saw a similar formula y = mx + b. Jul 13 1:45 PM Motion at onstant cceleration From these two formulas we can find some analogies: Velocity v y (depended variable of x), v 0 b (intersection with vertical axis), t x (independent variable), a m ( slope of the graph the ratio between rise and run Δy/Δx). y = m x + b v = a t + v 0 (or v = v 0 + a t) Jul 13 1:45 PM 5

6 Motion at onstant cceleration The formula a =Δv/Δt shows that the value of acceleration is the same as the slope on a graph of velocity versus time. Jul 13 2:07 PM 6 The velocity as a function of time is presented by the graph. What is the acceleration? a = slope = Δv/Δt = (10 m/s 2 m/s)/40 s = 0.2 m/s 2 Jul 13 3:49 PM 6

7 7 The velocity as a function of time is presented by the graph. Find the acceleration. a = slope = Δv/Δt = (0 m/s 25 m/s)/10 s = 2.5 m/s 2 Jul 14 9:59 M Motion at onstant cceleration The acceleration graph as a function of time can be used to find the velocity of a moving object. When the acceleration is constant it can be shown on the graph as a straight horizontal line. Jul 14 10:07 M 7

8 Motion at onstant cceleration In order to find the change in velocity for a certain limit of time we need to calculate the area under the acceleration versus time graph. The change in velocity during first 12 seconds is equivalent to the shadowed area (4χ12 = 48). The change in velocity during first 12 seconds is 48 m/s. Jul 14 10:07 M 8 What is the velocity of the object at 5 s? 1 m/s 2 m/s 3 m/s 4 m/s 5 m/s Jul 19 1:46 PM 8

9 9 Which of the following statements is true? The object speeds up The object slows down The object moves with a constant velocity The object stays at rest The object is in free fall Jul 19 1:46 PM 10 The following graph shows acceleration as a function of time of a moving object. What is the change in velocity during first 10 seconds? Δv = area = (3χ10) = 30 m/s Jul 14 10:33 M 9

10 11 The graph represents the relationship between velocity and time for an object moving in a straight line. What is the traveled distance of the object at 9 s? 10 m 24 m 36 m 48 m 56 m Jul 19 1:46 PM 12 Which of the following is true? The object increases its velocity The object decreases its velocity The object s velocity stays unchanged The object stays at rest More information is required Jul 19 1:46 PM 10

11 13 What is the initial position of the object? 2 m 4 m 6 m 8 m 10 m Jul 19 1:54 PM 14 What is the velocity of the object? 2 m/s 4 m/s 6 m/s 8 m/s 10 m/s Jul 19 1:54 PM 11

12 15 What is the initial position of the object? 2 m 4 m 6 m 8 m 10 m Jul 19 1:54 PM 16 The graph represents the position as a function of time of a moving object. What is the velocity of the object? 5 m/s 5 m/s 10 m/s 10 m/s 0 m/s Jul 19 1:54 PM 12

13 Free Fall ll unsupported objects fall towards the earth with the same acceleration. We call this acceleration the "acceleration due to gravity" and it is denoted by g. g = 9.8 m/s 2 Keep in mind, LL objects accelerate towards the earth at the same rate. g is a constant! Sep 15 10:22 M What It stops happens momentarily. at the top? v = 0 g = 9.8 m/s 2 It slows down. What happens when it (negative acceleration) goes up? g = 9.8 m/s 2 It speeds up What happens when it (negative goes down? acceleration) g = 9.8 m/s 2 n object is thrown upward with initial velocity, v o It What returns happens with when its it original lands? velocity. Sep 22 1:01 PM 13

14 It stops momentarily. v = 0 g = 9.8 m/s 2 It slows down. (negative acceleration) g = 9.8 m/s 2 It speeds up. (negative acceleration) g = 9.8 m/s 2 n object is thrown upward with initial velocity, v o It returns with its original velocity. Jun 9 11:13 PM On the way up: v a t = 3 s On the way down: v 0 v 1 a t = 0 s v 2 a t = 2 s a v 2 v 1 a t = 1 s v 1 v 2 t = 1 s a v v 2 t = 2 s v 0 a v 1 t = 0 s v t = 3 s Jun 30 11:38 M 14

15 For any object thrown straight up into the air, what does the velocity vs time graph look like? v (m/s) n object is thrown upward with initial velocity, v o It stops momentarily. v = 0 g = 9.8 m/s 2 t (s) It returns with its original velocity but in the opposite direction. Jun 30 12:07 PM 17 n object moves with a constant acceleration of 5 m/s 2. Which of the following statements is true? The object s velocity stays the same The object moves 5 m each second The object s acceleration increases by 5 m/s2 each second The object s acceleration decreases by 5 m/s2 each second the object s velocity increases by 5 m/s each second Jul 19 2:02 PM 15

16 18 truck travels east with an increasing velocity. Which of the following is the correct direction of the car s acceleration? Jul 19 2:02 PM 19 car and a delivery truck both start from rest and accelerate at the same rate. However, the car accelerates for twice the amount of time as the truck. What is the final speed of the car compared to the truck? Half as much The same Twice as much Four times as much One quarter as much Jul 20 9:46 M 16

17 20 car and a delivery truck both start from rest and accelerate at the same rate. However, the car accelerates for twice the amount of time as the truck. What is the traveled distance of the car compared to the truck? Half as much The same Twice as much Four times as much One quarter as much Jul 20 9:46 M 21 modern car can develop an acceleration four times greater than an antique car like Lanchester If they accelerate over the same distance, what would be the velocity of the modern car compared to the antique car? Half as much The same Twice as much Four times as much One quarter as much Jul 20 9:46 M 17

18 22 n object is released from rest and falls in the absence of air resistance. Which of the following is true about its motion? Its acceleration is zero Its acceleration is constant Its velocity is constant Its acceleration is increasing Its velocity is decreasing Jul 20 9:46 M 23 ball is thrown straight up from point it reaches a maximum height at point and falls back to point. Which of the following is true about the direction of the ball s velocity and acceleration between and? Jul 20 9:47 M 18

19 24 ball is thrown straight up from point it reaches a maximum height at point and falls back to point. Which of the following is true about the direction the ball s velocity and acceleration between and? Jul 20 9:47 M 25 football, a hockey puck, and a tennis ball all fall down in the absence of air resistance. Which of the following is true about their acceleration? The acceleration of the football is greater than the other two The acceleration of the hockey puck is greater than the other two The acceleration of the tennis ball is greater than the other two They all fall down with the same constant acceleration More information is required Jul 20 9:47 M 19

20 26 package is dropped from an air balloon two times. In the first trial the distance between the balloon and the surface is H and in the second trial 4H. ompare the time it takes for the package to reach the surface in the second trial to that in the first trial? The time in the second trial is four times greater The time in the second trial is two times greater The time the same in both trials because it doesn t depend on height The time in the second trial is four times less The time in the second trial is two times less Jul 20 9:47 M 27 Two baseballs are thrown from the roof of a house with the same initial speed, one is thrown up, and the other is down. ompare the speeds of the baseballs just before they hit the ground. The one thrown up moves faster because the initial velocity is up The one thrown down moves faster because the initial velocity is down They both move with the same speed The one thrown up moves faster because it has greater acceleration The one thrown down moves faster because it has greater acceleration Jul 20 9:47 M 20

21 28 n archer practicing with an arrow bow shoots an arrow straight up two times. The first time the initial speed is v0 and second time he increases the initial sped to 4v0. How would you compare the maximum height in the first trial to that in the first trial? Two times greater Four times greater ight times greater Sixteen times greater The same Jul 20 10:22 M 29 n acorn falls from an oak tree. You note that it takes 2.5 seconds to hit the ground. How fast was it going when it hit the ground? v = v o + at v=0 (9.8)2.5 v= 24.5 m/s Sep 15 10:20 M 21

22 30 n arrow is fired into the air and it reaches its highest point 3 seconds later. What was its velocity when it was fired? v = v o + at 0 = v o + 9.8(3) v o = 29.4m/s Sep 15 1:42 PM 31 You accelerate from 20m/s to 60m/s while traveling a distance of 200m; what was your acceleration? v 2 = vo 2 + 2aΔx 60 2 = a(200) a = 8m/s 2 Sep 6 10:41 PM 22

23 32 dropped ball falls 8.0m; what is its final velocity? v 2 = v o 2 + 2aΔy v 2 = 2aΔy v 2 = 2(9.8)8 v = 12.5m/s Sep 6 10:41 PM 33 ball with an initial velocity of 25m/s is subject to an acceleration of 9.8 m/s 2 ; how high does it go before coming to a momentary stop? v 2 = v o 2 + 2aΔy 0 = ( 9.8)Δy y = 31.9m Sep 6 10:41 PM 23