Describing motion (along a line) a.k.a. ‘the kinematics of linear motion’ Learning outcomes • define speed and acceleration, instantaneous and average values • explain the difference between relevant scalar and vector quantities • apply Galilean relativity to motions in inertial frames of reference • present a historical ‘thought experiment’ to illustrate physics thinking • establish concepts qualitatively (using proportional reasoning) before introducing quantitative relationships (equations) • choose contexts for teaching kinematics that motivate student learning • understand basic algebra and use it to rearrange kinematic equations • draw and interpret graphs of position, velocity, acceleration • translate information about uniform motions between words, pictures, graphs and equations • begin to develop a strategy for solving quantitative problems Starting points Misconceptions: Heavier objects are commonly thought to fall faster than lighter objects. Teaching challenges: • Concepts: Some students fail to grasp the distinction between velocity and acceleration – to them it’s simply ‘motion’. Acceleration is not simple idea: it is the rate of change of velocity, and velocity itself is the rate a change of distance (making acceleration the rate of change of a rate of change). • Graphs: Most students have difficulty with drawing and interpreting graphs representing motion (distinguishing s - t graphs from v - t graphs; appreciating significance of area under a v - t graph, of gradients of s - t and v - t graphs). • Equations: Students need help understanding that some equations constitute definitions and that other equations apply only when there is constant acceleration. Kinematics – describing motion Object is treated as a particle (a point-like concentration of matter that has no size, no shape and no internal structure). Questions to ask: • Where is the particle? • How fast is it moving? • How rapidly is it speeding up or slowing down? This is modelling. Restricted to motion along a line. Contexts In pairs: List other examples of real motion that might be modelled as a particle moving along a line. • Include some examples that can motivate students. Uniform motion Galileo (1638) Dialogue concerning two new sciences Definition: By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal. Galileo’s Two new sciences Axioms I The distance traversed during a _______ interval of time is greater than the distance travelled during a _______ interval of time. II The time required to traverse a _______ distance is longer than the time required for a _______ distance. III Over the same time interval, the distance traversed at a greater speed is _______ than the distance traversed at a ______ speed. IV The speed required to traverse a longer distance is greater than that required to traverse a ________ distance during the same time interval. Galileo’s Two new sciences Theorems I If a moving particle, carried at a constant speed, traverses two distances, the time intervals required are to each other in the ratio of these distances. II If a moving particle traverses two distances in equal intervals of time, these distances will bear to each other the same ratio as the speeds. And conversely, if the distances are as the speeds, then the times … III In the case of unequal speeds, the time intervals required to traverse a given space are to each other inversely as the speeds. We say … distance travelled speed time taken In symbols, s v t Other essential ingredients • a coordinate system • units: metres, seconds • scalar or vector? – distance, displacement – speed, velocity Measuring distances & times For class experiments & demonstrations • metre rules and stopwatches • ticker timers • light gates • sonic (ultrasound) sensor • video capture Discuss in small groups: What do you use? Graphical representation Uniform motion a) List the objects below in order of increasing speed. b) Which of the objects have positive velocity? c) List the objects in order of increasing velocity. Ticker timers Running on mains, they make 50 ticks each second. Time between ticks is therefore 1/50 s = 0.02 s Area under a v-t graph A car is driven along a straight road. The graph shows how the velocity of the car changes from the moment the driver sees a very slow moving queue of traffic ahead. Use the graph to calculate the distance the car travels while it is slowing down. Show clearly how you work out your answer. Finding an average speed 1 If you are not already familiar with ticker timers, first do the experiment Using the ticker-timer to measure time 2 Do one of these two experiments. Timing a trolley on a slope Pupil speed Naturally accelerated motion Aristotle: objects fall at constant speed; the more massive, the faster they fall. Galileo’s thought experiment. (Ignoring air resistance) All objects fall the same way, getting faster and faster. • a dramatic experimental test of this idea. http://www.physics.ucla.edu/demoweb/demomanual/mechanics/gra vitational_acceleration/guinea_and_feather_tube.html Free fall Galileo’s findings, modelled with chains. If the time of fall is twice as long, how much further does an object fall? • the v–t graph gives the answer. So what happens when an object is thrown vertically upwards? Graphical representation Constant acceleration v u a t units: km/h/s, m/s/s or m/s2 Walking the graph Using an ultrasound sensor to generate graphs. Every picture tells a story. An equation from the graph Equations of uniform motion Two definitions: s v t v u a t Four relationships derived from these: v u at uv s t 2 1 2 s ut at 2 v 2 u 2 2as Solving quantitative problems A standard approach 1. Write down what you know, using conventional symbols. 2. Decide what equation to use and write it down. 3. Re-arrange the equation to make the unknown its subject. 4. Substitute values and find the unknown quantity. 5. Write answer to correct number of significant figures, with units. Using equations of motion A worked example Some problems to solve Experiments about motion In fours: do a few of these experiments: Compensating for friction Investigating free fall with a light gate Measurement of g using an electronic timer Finding average acceleration with a ticker-timer Measurement of acceleration using light gates Building a reaction tester [from the Practical Physics website] Using simulation software Physlets: http://physics.bu.edu/~duffy/classroom.html PhET: http://phet.colorado.edu/simulations/ Video analysis: Multimedia Motion II, from Cambridge Science Media Discuss, in small groups: What do you use? Geogebra exercises Distance - time simulator Velocity Distance Time 1 Velocity Distance Time 2 Galilean relativity … and the idea of an inertial frame of reference Endpoints In small groups: Review the main ideas. Identify anything that is not clear and clarify it by discussion. Individually: Decide what you need to do to consolidate any or all of this material.
