Slide 1 / 125 New Jersey Center for Teaching and Learning Progressive Science Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org Slide 2 / 125 Kinematics in Two Dimensions www.njctl.org Slide 3 / 125 Table of Contents Click on the topic to go to that section Kinematics in One Dimension (Review) · Adding Vectors in Two Dimensions · Basic Vector Operations · Vector Components · Projectile Motion ·
Slide 4 / 125 Kinematics in One Dimension Return to Table of Contents Slide 5 / 125 Review of 1-D Kinematics · Kinematics is the description of how objects move with respect to a defined reference frame. · Displacement is the change in position of an object. · Average 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. · Average acceleration is the change in velocity divided by the time. Slide 6 / 125 Review of 1-D 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 + ½ at 2 v 2 = v o2 + 2 a(x - x o ) v = v + v o 2
Slide 7 / 125 1 A snapshot of three racing cars is shown on the diagram. All three cars start the race at the same time, at the same place and move along a straight track. As they approach the finish line, which car has the lowest average speed? Car I A Car II B Answer Car III C All three cars have the same average speed D http:/ / njc.tl/ 3m Slide 7 (Answer) / 125 1 A snapshot of three racing cars is shown on the diagram. All three cars start the race at the same time, at the same place and move along a straight track. As they approach the finish line, which car has the lowest average speed? Car I A C Car II B Answer Car III C All three cars have the same average speed D [This object is a pull tab] http:/ / njc.tl/ 3m Slide 8 / 125 2 A 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 A B Twice as much Answer C Four times as much D One quarter as much
Slide 8 (Answer) / 125 2 A 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 A B Twice as much B Answer C Four times as much D One quarter as much [This object is a pull tab] Slide 9 / 125 3 A 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 A The same B Answer Twice as much C Four times as much D Slide 9 (Answer) / 125 3 A 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 A D The same B Answer Twice as much C Four times as much D [This object is a pull tab]
Slide 10 / 125 4 A modern car can develop an acceleration four times greater than an antique car like “Lanchester 1800”. If they accelerate over the same distance, what would be the velocity of the modern car compared to the antique car? Half as much A The same B Answer Twice as much C Four times as much D Slide 10 (Answer) / 125 4 A modern car can develop an acceleration four times greater than an antique car like “Lanchester 1800”. If they accelerate over the same distance, what would be the velocity of the modern car compared to the antique car? Half as much A The same B C Answer Twice as much C Four times as much D [This object is a pull tab] Slide 11 / 125 Graphing Motion at Constant Acceleration In physics there is another approach in addition to algebraic which is called graphical analysis. The formula v = v 0 + at can be interpreted by the graph. We just need to recall our memory from math classes where we already saw a similar formula y = mx + b. From these two formulas we can make some analogies: v # y (dependent 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).
Slide 12 / 125 Motion at Constant Acceleration Below we can find the geometric explanation to the acceleration a =Δv/Δt. If slope is equal to: m = Δy/Δx Then consider a graph with velocity on the y-axis and time on the x-axis. What is the slope for the graph on the right? Slide 13 / 125 Motion at Constant Acceleration The graph on the right has a slope of Δv/Δt, which is equal to acceleration. Therefore, the slope of a velocity vs. time graph is equal to acceleration. (slope) (slope of velocity vs. time) y =Δy/Δx a =Δv/Δt Slide 14 / 125 5 The velocity as a function of time is presented by the graph. What is the acceleration? Answer http:/ / njc.tl/ 3n
Slide 14 (Answer) / 125 5 The velocity as a function of time is presented by the graph. What is the acceleration? a = slope = Δ v/ Δ t Answer a =(10 m/s -2 m/s)/40 s a= 0.2 m/s 2 [This object is a pull tab] http:/ / njc.tl/ 3n Slide 15 / 125 6 The velocity as a function of time is presented by the graph. Find the acceleration. Answer http:/ / njc.tl/ 3o Slide 15 (Answer) / 125 6 The velocity as a function of time is presented by the graph. Find the acceleration. a = slope = Δ v/ Δ t Answer a = (0 m/s - 25 m/s)/10 s a = -2.5 m/s 2 [This object is a pull tab] http:/ / njc.tl/ 3o
Slide 16 / 125 Motion at Constant Acceleration 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. Slide 17 / 125 Motion at Constant Acceleration 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 (4x12 = 48). The change in velocity during first 12 seconds is 48 m/s. Slide 18 / 125 7 Which of the following statements is true? A The object slows down Answer B The object moves with a constant velocity C The object stays at rest D The object is in free fall http:/ / njc.tl/ 3q
Slide 18 (Answer) / 125 7 Which of the following statements is true? A The object slows down Answer B B The object moves with a constant velocity C The object stays at rest [This object is a pull tab] D The object is in free fall http:/ / njc.tl/ 3q Slide 19 / 125 8 The following graph shows acceleration as a function of time of a moving object. What is the change in velocity during first 10 seconds? Answer http:/ / njc.tl/ 3r Slide 19 (Answer) / 125 8 The following graph shows acceleration as a function of time of a moving object. What is the change in velocity during first 10 seconds? Δ v = area Answer = (3 χ 10) = 30 m/s [This object is a pull tab] http:/ / njc.tl/ 3r
Slide 20 / 125 Analyzing Position vs Time Graphs Recall earlier in this unit that slope was used to describe motion. x The slope in a position vs. time (m) graph is Δx/Δt, which is equal to Δx velocity. v = Δx/Δt Δt Therefore, slope is equal to velocity on a position vs. time t (s) graph. http:/ / njc.tl/ 2t Slide 21 / 125 Analyzing Position vs Time Graphs A positive slope is a positive velocity, a negative slope is a negative velocity, and a slope of zero means zero velocity. positive slope negative slope zero slope v > 0 v < 0 v = 0 x x x (m) (m) (m) t (s) t (s) t (s) A positive velocity means moving in the positive direction, a negative velocity means moving in the negative direction, and zero velocity means not moving at all. http:/ / njc.tl/ 2t Slide 22 / 125 9 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 A 24 m B C 36 m 48 m D Answer http:/ / njc.tl/ 3s
Slide 22 (Answer) / 125 9 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 A B 24 m 36 m C 48 m D Answer C [This object is a pull tab] http:/ / njc.tl/ 3s Slide 23 / 125 10 Which of the following is true? The object increases its velocity A The object decreases its velocity B Answer C The object’s velocity stays unchanged The object stays at rest D http:/ / njc.tl/ 3t Slide 23 (Answer) / 125 10 Which of the following is true? A The object increases its velocity The object decreases its velocity B Answer The object’s velocity stays unchanged C C The object stays at rest D [This object is a pull tab] http:/ / njc.tl/ 3t
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