One Dimensional Motion Section 1 Displacement and Velocity Chapter 2 One Dimensional Motion To simplify the concept of motion, we will first consider motion that takes place in one direction. One example is the motion of a commuter train on a straight track. To measure motion, you must choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time.

displacement = final position – initial position Section 1 Displacement and Velocity Chapter 2 Displacement Displacement is a change in position. Displacement is not always equal to the distance traveled. The SI unit of displacement is the meter, m. Dx = xf – xi displacement = final position – initial position

Positive and Negative Displacements Section 1 Displacement and Velocity Chapter 2 Positive and Negative Displacements

Chapter 2 Average Velocity Section 1 Displacement and Velocity Chapter 2 Average Velocity Average velocity is the total displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per second, abbreviated as m/s.

Chapter 2 Velocity and Speed Section 1 Displacement and Velocity Chapter 2 Velocity and Speed Velocity describes motion with both a direction and a numerical value (a magnitude). Speed has no direction, only magnitude. Average speed is equal to the total distance traveled divided by the time interval.

Interpreting Velocity Graphically Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically For any position-time graph, we can determine the average velocity by drawing a straight line between any two points on the graph. If the velocity is constant, the graph of position versus time is a straight line. The slope indicates the velocity. Object 1: positive slope = positive velocity Object 2: zero slope= zero velocity Object 3: negative slope = negative velocity

Interpreting Velocity Graphically, continued Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically, continued The instantaneous velocity is the velocity of an object at some instant or at a specific point in the object’s path. The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position-versus-time graph.

Chapter 2 Changes in Velocity Section 2 Acceleration Changes in Velocity Acceleration is the rate at which velocity changes over time. An object accelerates if its speed, direction, or both change. Acceleration has direction and magnitude. Thus, acceleration is a vector quantity.

Changes in Velocity, continued Chapter 2 Section 2 Acceleration Changes in Velocity, continued Consider a train moving to the right, so that the displacement and the velocity are positive. The slope of the velocity-time graph is the average acceleration. When the velocity in the positive direction is increasing, the acceleration is positive, as at A. When the velocity is constant, there is no acceleration, as at B. When the velocity in the positive direction is decreasing, the acceleration is negative, as at C.

Relationship Between Acceleration and Velocity Chapter 2 Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

Relationship Between Velocity and Acceleration Chapter 2 Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer)

Relationship Between Velocity and Acceleration Chapter 2 Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter)

Kinematic Equations a = Δ v = vf - vi ti = 0 Δ t tf - ti Therefore vf = vi + at

Kinematic Equations Vavg = Δx t Δx = vavgt = vi + vf t 2 Δx = ½ (vi + vf) t But vf = vi + at Δx = ½ (vi + vi + at) t Δx = vi t + ½ at2

Kinematic Equations Δx = ½ (vi + vf) t We can develop an equation without t vf = vi + at t = vf - vi (sub. into equation above for ΔX) a vf2 = vi2 + 2aΔx

x a 2 v at 1 t D + = ÷ ø ö ç è æ Kinematic Equations Summary Uniform Acceleration equations of motion x a 2 v at 1 t i f average D + = ÷ ø ö ç è æ

Velocity and Acceleration Chapter 2 Section 2 Acceleration Velocity and Acceleration

Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth’s surface fall with a constant acceleration Galileo originated our present ideas about free fall from his inclined planes The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity Free Fall Acceleration due to Gravity Symbolized by g g = 9.8 m/s² g is always directed downward toward the center of the earth

Free Fall -- an object dropped Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical vi= 0 a = g

Free Fall – an object thrown downward a = g Initial velocity  0 With upward being positive, initial velocity will be negative

Free Fall – object thrown upward Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a = g everywhere in the motion g is always downward, negative v = 0