Lesson : Kinematics (Sections.-.5) Chapter Motion Along a Line In order to specify a position, it is necessary to choose an origin. We talk about the football field is 00 yards from goal line to goal line, it is 35 miles until we arrie at grandma s, and so on. When motion occurs along a line, we customarily choose the -ais along the motion. An eception occurs when things are moing up and down. In those cases the motion occurs in the y direction. We define the displacement as the change in position: f i We will use the (delta) notation often. It denotes the change in whateer follows. By definition, the change is the difference in the final and initial alues. Displacement is different from distance. The upper displacement is 8 m and the lower displacement is 9 m. What is the total displacement? What is the total distance? The displacement depends only on the beginning and ending points, not the path taken. The total displacement for multiple displacements is the sum of the indiidual displacements. How does that work for the problem aboe? Een though positions depend on the location of the origin, displacements do not. When a displacement occurs in a time t, we define the aerage elocity as a, t Lesson, page
Lesson : Kinematics (Sections.-.5) Since the time interal is always positie (why?), the aerage elocity is in the same direction as the displacement. The s do not cancel in the definition. The operates on the ariable, it does not multiply it. The aerage speed is different from the aerage elocity. In the problem aboe, suppose the motion occurs in 0 s. Aerage elocity is 9 m/0 s = 0.45 m/s. Aerage speed is 7 m/0 s =.35 m/s. If you run around a track, your aerage elocity will be zero if the starting and ending points coincide. What are the units of elocity? Can the aerage speed be greater than the aerage elocity? The instantaneous elocity is the elocity of the object at some moment in time. There can be a big difference between aerage elocity and instantaneous elocity. Just consider any trip around Gainesille. The instantaneous elocity is the limit of taking small displacements oer smaller time interals: lim t0 t Okay, calculus people. What do we call this? As the time interal shrinks, the slope of the chord approaches a constant alue and becomes a tangent line. The instantaneous elocity is the slope of a position s. time graph. Lesson, page
Lesson : Kinematics (Sections.-.5) Let s talk about the graph aboe. From now on, we drop the instantaneous and just call it elocity. If we look at a elocity s. time graph, we can find the displacement. For constant elocity, the graph is How do we know the elocity is constant? If the elocity is constant, the aerage and instantaneous elocities are the same, a, t t The displacement is the area under the cure in a elocity s. time graph. Read the warning in page 33. Lesson, page 3
Lesson : Kinematics (Sections.-.5) If the elocity changes, it can be harder to find : Okay calculus people, what do we call this? In a similar way, we can talk about the rate of change of elocity. The rate of change of elocity is called the acceleration. The aerage acceleration is a a, t What are the units of acceleration? The instantaneous acceleration (or just the acceleration) is a lim t0 t The acceleration is the slope of a elocity s. time graph. The area under an acceleration s. time graph is the elocity. Which way is the object heading in the graph aboe? Is it going faster or slower? Lesson, page 4
Lesson : Kinematics (Sections.-.5) There can be confusion when we talk about acceleration. For eample, what to you push on the make your car trael with constant speed? We will not use the word deceleration in this class. Rather we will concern ourseles with the direction of the acceleration and elocity. Here are four possibilities: Situation in positie direction, a in positie direction in positie direction, a in negatie direction in negatie direction, a in positie direction in negatie direction, a in negatie direction Result Magnitude of elocity increases Magnitude of elocity decreases Magnitude of elocity decreases Magnitude of elocity increases To summarize: When the elocity and acceleration point in the same direction, the speed increases. When they point in opposite directions, the speed decreases. In straight line motion, the acceleration is always in the same direction as the elocity, in the direction opposite to the elocity, or zero. (page 36) To summarize the relationships between the graphs: slope slope s. t graph s. t graph a s. t graph area area Our first set of formulas: The equations of uniformly accelerated motion. I will follow the deriation in the book. (See pages 39-4) If the acceleration is constant, the aerage acceleration is equal to the instantaneous acceleration. a aa, a t t Our first equation is f i a t It is just a slight rearrangement of the definition of acceleration. Lesson, page 5
Lesson : Kinematics (Sections.-.5) The second equation is found from the diagrams. Since the acceleration is constant, the aerage elocity is the aerage of the initial and final elocities: a (, f i The constant acceleration shows up as a constant slope in the elocity s. time graph. Using this with the definition of aerage elocity, ) a, t a, ( t f i ( f ) t i ) t Now let us substitute for f in the aboe equation. ( t i f i ) t a ( t) (( i a t) i ) t We could hae diided the two equations, and found ( f i )( f f f i i i ( ) a a f a t i ) t Our four formulas of uniformly accelerated motion, Equations -9, -, -, and -3, are Lesson, page 6
Lesson : Kinematics (Sections.-.5) f i a t The difference in elocity is related to the acceleration and how long the acceleration acts. ( ) t The displacement is the aerage elocity times the time. f i t i ) a ( t The displacement is the sum of the displacement due to the initial speed and the displacement due to the acceleration f i a The difference is the squares of the elocities is related to the acceleration and the displacement. While this one is hard to see, it will hae HUGE consequences later. Problem.5 A train is traeling along a straight, leel track at 6.8 m/s. Suddenly the engineer sees a truck stalled on the tracks 84 m ahead. If the maimum possible braking acceleration has magnitude.5 m/s, can the train be stopped in time? Solution: If the train in not going to hit the truck, it must stop before traeling the 84 m with the maimum acceleration of.5 m/s. Choose the equation that does not hae the time, f i f f a i a i 6.8 m/s.5 m/s 84m.6 m/s a Why is the acceleration negatie? What acceleration was needed to stop the train in time? f a i a a f f i i 0 6.8 m/s 84m.95 m/s Lesson, page 7
Lesson : Kinematics (Sections.-.5) When does the collision occur? t i a t To sole this equation for the time, we will need to use the quadratic formula. Usually I think you should put in the numerical alues at the end, but when dealing with the quadratic, you should plug in the numbers at the beginning. it a t 6.8 m/s t.5 m/s 84m t To simplify the equation more I will drop the units as well, 0.76 84 t 6.8t 84 0 Recall that the solution to the general quadratic equation is a b c 0 6.8t.5 t b b 4ac a Matching the two equations, we hae the solution t 6.8 6.8 40.7684 0.76 9.34s, 5.9s Which answer is correct? Are both correct? A much easier way is to use the definition of acceleration, f i at f t a i.6 m/s 6.8 m/s.5 m/s 9.34s The correct root is the 9.34 s solution. What is the meaning of the second root? If there was no truck to collide with and the train continued its acceleration een after it stopped and began to Lesson, page 8
Lesson : Kinematics (Sections.-.5) back up, it would return to a the point 84 m away after 5.9 seconds. But since there is a truck to collide with, the second root is not possible. We call it unphysical. Motion diagrams can be helpful. The pictures are taken at equal time interals. Which is moing with constant speed? Which has an increasing elocity? A decreasing elocity? (I know the answer is in the picture!) Lesson, page 9