Chapter 1 Driving the Roads SECTION 5 Negative Acceleration: Braking Your Automobile Section Overview speed or acceleration is the change in the speed of an object in a given time period. Acceleration, This section addresses the distance traveled avt=Δ Δ (Δv represents the change in velocity that during negative acceleration as a function of the is measured by recording velocity at two different initial speed. Students continue the cart-stopping periods). The initial velocity is often indicated as v1 investigation by analyzing the problem of “slowing and fi nal velocity as v2. In this section, the students down” a cart. Students collect data by setting up will be solving for the distance required to stop, an experiment in which a toy cart is rolled down a while traveling at different velocities. ramp. The initial speed of the cart versus its braking distance is plotted on a graph. The change in speed Plotting speed vs. time, results in a straight line, of the cart as it comes to a stop is explained by with the slope of that line indicating acceleration. using the term negative acceleration. The effect of increasing the initial speed is recorded as data and then plotted on a speed versus time graph. Students study the similarities and differences in their data to ) fi nd the relationship of initial speed to the braking v distance. Braking distance is fi nally defi ned and determined to be proportional to the initial velocity speed ( squared. time (t) Background Information avt=Δ Δ The physics of stopping a cart involves reaction time while moving with constant or uniform motion, and The unit for v is m/s and for t is s. braking time while braking with changing speed or Therefore, nonuniform motion. If students plot the data for Δv distance versus time, for nonuniform motion, they a = Δt get a parabola. and with units, ) m/s 2 d a ==m/s s However, when looking at braking distances, distance ( distance traveled is the primary concern. For example, knowing the distance a plane requires time (t) to land is important when building a runway. It is also important to know how far an automobile will Unfortunately, reading and making predictions from travel while braking. Therefore, the distance that a parabola is very diffi cult. Students would require is required when an automobile is slowing down calculus to make sense of such a graph. Changing 142 Active Physics AAPTE_C1_S3_7.inddPTE_C1_S3_7.indd 142142 77/28/09/28/09 33:52:20:52:20 PMPM Section 5 Overview is considered. From a speed-time graph, fi nding the distance as it relates to changes in the initial area under the graph will give the total distance velocity. If the velocity doubles, the braking distance covered. quadruples. If the velocity triples, the braking distance is nine times as great. If the velocity CHAPTER 1 quadruples, the braking distance is sixteen times longer. Similarly, if the velocity is halved, the braking ) v distance is quartered. This equation can be derived from the defi nition of speed ( velocity and acceleration. Δv vv− time (t) a ==fi Δt t The area under the graph above is expressed as vatv=+ =×1 fi Area 2 bh. Using the units from the graph, dvt= 2. Going back to the original equation of Since = acceleration, avt, and rearranging to solve for t vvat==−0 and tva= dvt= 2 fi gives . Combining , and 222 1 2 2 va= ttataa= ( )22= d. gives dva= 2 , which is the distance an object i 2 travels proportional to the square of its speed. In addition, this section pays particular attention Therefore, when the speed the object is traveling is to the concept of negative acceleration, both as doubled, the distance traveled is quadrupled. an indication of decreasing speed, and also in the context of the vector nature of acceleration. Rather EXAMPLE than use the term “deceleration,” the term negative acceleration is used. An object is traveling at 3 m/s with a negative 2 acceleration of 2 m/s , what is the distance traveled? To clarify the physicist’s use of the term negative acceleration, it is necessary to deal with the vector Negative acceleration means that the car is slowing nature of acceleration. Acceleration may occur in down. Therefore, any direction. If one direction, for example, east, v2 is arbitrarily chosen to be positive, an increasing d = 2a speed in the easterly direction would be a positive or , acceleration, while a decreasing speed in that 2 (3 m/s) direction would be considered negative. An increase d = = 225. m. 2 in speed toward the west (the negative direction) 22( m/s ) would then be a negative acceleration. If the speed increases, it becomes more negative. If the speed Now suppose the speed is doubled to 6 m/s, 2 decreases in the negative direction, it gradually (6 m/s) becomes less negative and ultimately changes to a d = = 9 m. 22( m/s2 ) positive velocity, indicating a positive acceleration. The kinematics of the relationship which relates As an enrichment activity, the students can run the braking distance and speed is vad2 = 2 . same investigation on different surfaces. They could then draw analogies to stopping on different road In this section, expect students to recognize that surfaces and how they would drive under those v2 is proportional to distance. This can be best circumstances. An example might be running the understood by looking at changes in the braking cart on sandpaper, wet fl oor, dirt surface, etc. 143 Active Physics AAPTE_C1_S3_7.inddPTE_C1_S3_7.indd 143143 77/28/09/28/09 33:52:21:52:21 PMPM Chapter 1 Driving the Roads Crucial Physics • Braking distance is dependent on the negative acceleration of the car (brakes, road surface) and reaction time. 2 ° dv= 2 a. ° Tripling the speed requires nine times the braking distance. • Positive acceleration and negative acceleration are not associated uniquely with speeding up or slowing down. ° One determines if an object speeds up or slows down by considering both the direction of the initial velocity and the direction of the acceleration. Learning Outcomes Location in the Section Evidence of Understanding Plan and carry out an Investigate Students plan and carry out an experiment to see how the experiment to relate braking Steps 2, 3, and 4 initial speed of a cart affects its braking distance by varying distance to initial speed. the speed of a cart traveling down a ramp. Determine braking distance. Investigate Students use data to determine the braking distance Step 5 of a cart. Examine accelerated motion. Investigate Students study graphs to see how varying the initial speed Steps 5–8 changes the braking distance. NOTES 144 Active Physics AAPTE_C1_S3_7.inddPTE_C1_S3_7.indd 144144 77/28/09/28/09 33:52:21:52:21 PMPM Section 5 Overview Section 5 Materials, Preparation, and Safety Materials and Equipment Time Requirement CHAPTER 1 In order for students to run 10 good trials, after PLAN A proper instruction, they would need at least 45 min. Group Materials and Equipment Class (If there is only one set of equipment for the entire (4 students) class, then adjust the appropriate amount of time Meter stick, wood 1 per group accordingly.) Allow for one more class period to plot Ring stand, large 1 per group the data and analyze the graph. Rod, aluminum, 12 in. (length) x 3/8 in. (diameter) (to act as 1 per group Teacher Preparation crossarm) Holder, right angle (to act as • Determine the speed-measuring equipment to be 1 per group crossarm) used. If a velocimeter or a motion detector is not Inclined plane ramp for lab cart 1 per group available, you may use a camcorder, monitor, and Dynamics cart 1 per group a VCR with stop, forward, and advance to show a clear picture of each frozen frame. Practice using the Scissors 1 per group camcorder if you are not familiar with it. Be ready Extension clamp 2 per group to advise students on how to use the freeze frame Velocimeter 1 per group advance capability of the tape player or camcorder. Index cards, pkg 100 1 per class Secure a suitable scale behind the path of the cart so Folder, fi le 10 per class that its position is easily observed on the screen. Paper, graph, pkg of 50 1 per class • If no method is available to measure the speed at *Additional items needed not supplied the bottom of the ramp, measure the time of descent PLAN B from rest to the bottom of the ramp. Calculate the average speed on the ramp and double it to Group Materials and Equipment Class (4 students) get the speed at the bottom. While this method avoids much apparatus, it requires them to trust Meter stick, wood 1 per class and understand the mathematics. They are more Ring stand, large 1 per class likely to trust the technology that gives a direct Rod, aluminum, 12 in. (length) measurement. Be sure that the transition from the x 3/8 in. (diameter) (to act as 1 per class ramp to the fl oor is smooth. A fi le folder taped to crossarm) the inclined plane and the surface will suffi ce. Holder, right angle (to act as 1 per class • If the Investigate is to be performed on the fl oor, crossarm) inform students the day prior so they may wear Inclined plane ramp for lab cart 1 per class appropriate clothing. Dynamics cart 1 per class Scissors 1 per class Safety Requirements Extension clamp 2 per class • If the Investigate is done on the fl oor, students Velocimeter 1 per class should be aware of the equipment so they do not Index cards, pkg 100 1 per class walk into it.