Chapter 2 One-Dimensional Kinematics
Units of Chapter 2 Position, Distance, and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration Motion with Constant Acceleration Applications of the Equations of Motion Freely Falling Objects
2-1 Position, Distance, and Displacement Before describing motion, you must set up a coordinate system define an origin and a positive direction.
2-1 Position, Distance, and Displacement The distance is the total length of travel; if you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi.
2-1 Position, Distance, and Displacement Displacement is the change in position. If you drive from your house to the grocery store and then to your friend s house, your displacement is 2.1 mi and the distance you have traveled is 10.7 mi.
2-2 Average Speed and Velocity The average speed is defined as the distance traveled divided by the time the trip took: Average speed = distance / elapsed time Is the average speed of the red car 40.0 mi/h, more than 40.0 mi/h, or less than 40.0 mi/h?
2-2 Average Speed and Velocity Average velocity = displacement / elapsed time If you return to your starting point, your average velocity is zero.
2-2 Average Speed and Velocity Graphical Interpretation of Average Velocity The same motion, plotted one-dimensionally and as an x-t graph:
2-3 Instantaneous Velocity Definition: (2-4) This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity.
2-3 Instantaneous Velocity This plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is tangent to the curve.
2-3 Instantaneous Velocity Graphical Interpretation of Average and Instantaneous Velocity
2-4 Acceleration Average acceleration: (2-5)
2-4 Acceleration Graphical Interpretation of Average and Instantaneous Acceleration:
2-4 Acceleration Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration:
2-5 Motion with Constant Acceleration If the acceleration is constant, the velocity changes linearly: Average velocity: (2-7)
2-5 Motion with Constant Acceleration Average velocity: (2-9) Position as a function of time: (2-10) (2-11) Velocity as a function of position: (2-12)
2-5 Motion with Constant Acceleration The relationship between position and time follows a characteristic curve.
2-5 Motion with Constant Acceleration
2-6 Applications of the Equations of Motion Hit the Brakes!
2-7 Freely Falling Objects Free fall is the motion of an object subject only to the influence of gravity. The acceleration due to gravity is a constant, g.
2-7 Freely Falling Objects An object falling in air is subject to air resistance (and therefore is not freely falling).
2-7 Freely Falling Objects Free fall from rest:
Summary of Chapter 2 Distance: total length of travel Displacement: change in position Average speed: distance / time Average velocity: displacement / time Instantaneous velocity: average velocity measured over an infinitesimally small time
Summary of Chapter 2 Instantaneous acceleration: average acceleration measured over an infinitesimally small time Average acceleration: change in velocity divided by change in time Deceleration: velocity and acceleration have opposite signs Constant acceleration: equations of motion relate position, velocity, acceleration, and time Freely falling objects: constant acceleration g = 9.81 m/s 2
Chapter 4 Two-Dimensional Kinematics
Units of Chapter 4 Motion in Two Dimensions Projectile Motion: Basic Equations Zero Launch Angle General Launch Angle Projectile Motion: Key Characteristics
4-1 Motion in Two Dimensions If velocity is constant, motion is along a straight line:
4-1 Motion in Two Dimensions Motion in the x- and y-directions should be solved separately:
4-2 Projectile Motion: Basic Equations Assumptions: ignore air resistance g = 9.81 m/s 2, downward ignore Earth s rotation If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is -9.81 m/s 2
4-2 Projectile Motion: Basic Equations The acceleration is independent of the direction of the velocity:
4-2 Projectile Motion: Basic Equations These, then, are the basic equations of projectile motion:
4-3 Zero Launch Angle Launch angle: direction of initial velocity with respect to horizontal
4-3 Zero Launch Angle In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x 0 = 0 and y 0 = h:
4-3 Zero Launch Angle This is the trajectory of a projectile launched horizontally:
4-3 Zero Launch Angle Eliminating t and solving for y as a function of x: This has the form y = a + bx 2, which is the equation of a parabola. The landing point can be found by setting y = 0 and solving for x:
4-4 General Launch Angle In general, v 0x = v 0 cos θ and v 0y = v 0 sin θ This gives the equations of motion:
Solution page 91
4-4 General Launch Angle Snapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s
4-5 Projectile Motion: Key Characteristics Range: the horizontal distance a projectile travels If the initial and final elevation are the same:
4-5 Projectile Motion: Key Characteristics The range is a maximum when θ = 45 :
4-5 Projectile Motion: Key Characteristics Symmetry in projectile motion:
Summary of Chapter 4 Components of motion in the x- and y- directions can be treated independently In projectile motion, the acceleration is g If the launch angle is zero, the initial velocity has only an x-component The path followed by a projectile is a parabola The range is the horizontal distance the projectile travels