Physics 1110 Fall 2004L5 - 1Lecture 5 1 September 2004 Announcements• Check the course web page for new information• Register your “clickers” via the link on the course web pageGraphical Representation of Uniform MotionWe continue our study of graphical representation by examining thesimple case of uniform motion. Recall that a motion diagram foruniform motion has equally spaced dots along a line, indicating thatthe velocity has constant direction and magnitude, and theacceleration is zero, as shown in the figure.Taking the time between each dot to be 1 second, we obtain thefollowing x vs t plot:Physics 1110 Fall 2004L5 - 2Note that the points fall along a straight line, shown in the figure toguide your eye. This is the signature of constant velocity!Furthermore, since this is a plot of uniform motion we know that thedisplacement x between each point is the same. Combining thiswith the equal time steps t, we can say mathematically, what isclear from the picture: the slope of the line x/t is a constant, and infact is the velocity in the x-direction vx. This is shown on the figurebelow.Before going further, there are a number of comments to make here.• Note that position being positive or negative tells you where theobject is relative to the origin, while positive or negativedisplacement tells you whether the change was in the positiveor negative x direction. Thus the sign of the displacement tellsyou the direction! This is generally true when we consider 1-dimensional vectors along a coordinate axis.• For the same reason, the velocity along the x-direction, vx,which is x/t, points in the positive or negative directiondepending on its sign + or -.• Thus it is quite possible to have a negative displacement fromtwo positive positions; it just means the slope is negative.Physics 1110 Fall 2004L5 - 3Instantaneous VelocityThe text gives a very clear description of the concept of theinstantaneous velocity and how it is related to the first derivative of afunction which describes x as a function of t (written x(t)). The point tostress here is that the observational definition of the instantaneousvelocity,vx= limt 0xt,implies a measurement procedure. If I try to measure thedisplacement x for shorter and shorter times t, eventually I reach aregime where the graph of x vs t looks like a straight line with the awell defined slope, even if the function has curvature at a largerscale. In other words, for smaller and smaller time steps thecalculation of x/t doesn’t change anymore. This is what “having aslope at a point on a curve” means (often shown as a line tangent tothe point on the curve). It is entertaining to note that while thisconcept is pure in the mathematics of calculus, nature does not allowsuch a procedure for infinitesimally small time steps. When thedistances and times get very small, quantum mechanics is the truetheory of motion, and we must refine these simple, classical ideas.Finally, the figure below shows the velocity plot vs time for theposition plot shown above. For uniform motion, the velocity isconstant, hence it is just a horizontal line.Physics 1110 Fall 2004L5 - 4The area between this straight line and the x-axis (between two giventimes) is in fact equal to the displacement between those times. Thisis a bit strange because we usually think of an area as having units ofdistance squared. But in this case we must use our graphical scalesto interpret the “area,” so for an area defined by a vertical side and ahorizontal side, we will get meter/sec times sec to give a result inmeters, the correct unit for a displacement.1D Motion with Constant AccelerationThe next case to investigate is that of constant acceleration. Since weare only considering 1D motion, this means there are only two cases:acceleration in the same direction as velocity and acceleration in theopposite direction of the velocity. Below are motion diagrams for thetwo cases. The first is for the case of a and v in the same direction;the second is the case of a and v in the opposite direction.These diagrams show (roughly) that the velocity changes at aconstant rate, i.e., a constant acceleration. The graph of velocity vstime is therefore a line with a constant slope, which is equal to theacceleration. To find the position plot, we use the principle discussedabove, that the area between the velocity line and the time axis givesPhysics 1110 Fall 2004L5 - 5the displacement. The figure below shows a case with constantpositive acceleration and an initial positive velocity of v0.We can use simple geometry to find the area, since it is the sum of arectangular area = v0t plus a triangular area of (1/2)(at)t. Fromthis we know that x = v0t + (1/2)a(t)2, which is a quadratic formula.This dependence on time gives a parabola.In the next lecture, we’ll continue to explore the case of constantacceleration, especially the most common example of such motion,that of free fall under the influence of planetary
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