Physics 1D03 - Lecture 21 Kinematics in One Dimension Displacement, velocity, acceleration Graphs A special case: constant acceleration Bodies in free fall Serway and Jewett Chapter 2

Physics 1D03 - Lecture 22 1-D motion can be described by scalars (real numbers with units) as functions of time: The sign (positive or negative) keeps track of direction (in 1-D). Algebraic relations involving position, velocity, and acceleration come from calculus. The same relations can be seen from graphs of position, velocity, and acceleration as functions of time. Position x(t) (displacement from the origin) Velocity v(t) (rate of change of position) Acceleration a(t) (rate of change of velocity)

Physics 1D03 - Lecture 23 Kinematics : the description of motion One dimension : motion along a straight line (e.g., the x-axis) Examples - sprinter running 100 meters in a straight line - ball falling straight down, and bouncing back up

Physics 1D03 - Lecture 24 position x as a function of time t Average velocity : (slope of the line) xx tt x1x1 x2x2 t1t1 t2t2 t x Displacement :

Physics 1D03 - Lecture 25 Instantaneous velocity is the average over an ‘infinitesimal’ time interval : v is the slope of the tangent to the x vs. t graph. Physically, v is the rate of change of x, hence dx/dt. tt x

Physics 1D03 - Lecture 26 Acceleration is the rate of change of velocity:

Physics 1D03 - Lecture 27 Graphs of x(t), v(t), a(t)

Physics 1D03 - Lecture 28 Graphs of x(t), v(t), a(t)

Physics 1D03 - Lecture 29 Quiz A rubber ball is dropped and bounces twice from the floor before it is caught. (Take x to be upwards, and x=0 at the floor.) At the highest point of the first bounce, v and a are: a) both nonzero b) one is zero, one is not zero c) both zero d) other (explain) Suggestion: Sketch graphs of x, v, a vs. time.

Physics 1D03 - Lecture 210 Quiz A particle (in one dimension) is initially moving. A few seconds later it has stopped (not moving). During that time interval: a)The particle’s average acceleration is positive b)The particle’s average acceleration is negative c)Not enough information to tell

Physics 1D03 - Lecture 211 Exercise: eliminate t or a to show that Caution: These assume acceleration is constant. These are sometimes convenient, but not necessary. They are valid only for constant acceleration. Use the definitions and derive A Special Case: Constant Acceleration

Physics 1D03 - Lecture 212 All objects in free fall move with constant downward acceleration, This was demonstrated by Galileo around 1600 A.D. “g” is called the “acceleration due to gravity” or the “gravitational field of the Earth”. Example: Free Fall. (“Free fall” means the only force is gravity; the motion can be in any direction).

Physics 1D03 - Lecture 213 The free-fall acceleration is the same for all objects; size and composition don’t matter. But: g varies slightly with location and height, about  0.03 m/s 2 over the surface of the Earth, and up to a few kilometers above if air resistance is significant, we don’t really have “free fall”.

Physics 1D03 - Lecture 214 Quiz A block is dropped from rest. It takes a time t 1 to fall the first third of the distance. How long does it take to fall the entire distance? a) t 1 b)3t 1 c)9t 1 d)None of the above

Physics 1D03 - Lecture 215 Quiz You throw a set of keys up to a window 4.9m above you. If the keys just make it to your friend on a balcony 1.0s later, what was their initial velocity ? a)40 m/s b)9.8 m/s c)4.5 m/s