Describing Motion: Kinematics in One Dimension Chapter 2 Describing Motion: Kinematics in One Dimension

Reference Frames and Displacement Any measurement of position, distance, or speed must be made with respect to a reference frame. For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.

Reference Frames and Displacement We make a distinction between distance and displacement. Displacement (blue line) is how far the object is from its starting point, regardless of how it got there. Distance traveled (dashed line) is measured along the actual path.

Reference Frames and Displacement The displacement is written: Displacement is positive. Displacement is negative.

Defining the important variables Kinematics is a way of describing the motion of objects without describing the causes. You can describe an object’s motion: In words Mathematically Pictorially Graphically No matter HOW we describe the motion, there are several KEY VARIABLES that we use. Symbol Variable Units t Time s a Acceleration m/s/s x or y Displacement m vo Initial velocity m/s v Final velocity g or ag Acceleration due to gravity

Average Velocity Speed: how far an object travels in a given time interval (2-1) Velocity includes directional information:

How far can a cyclist travel in 2 How far can a cyclist travel in 2.5 h along a straight road if her average velocity is 18km/hr?

Instantaneous Velocity The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short. (2-3) These graphs show (a) constant velocity and (b) varying velocity.

Acceleration Acceleration is the rate of change of velocity.

Acceleration Acceleration is a vector, although in one-dimensional motion we only need the sign. The previous image shows positive acceleration; here is negative acceleration:

Acceleration There is a difference between negative acceleration and deceleration: Negative acceleration is acceleration in the negative direction as defined by the coordinate system. Deceleration occurs when the acceleration is opposite in direction to the velocity. The magnitude of the velocity is decreasing.

A car accelerates along a straight road from rest to 75 km/h in 5. 0 s A car accelerates along a straight road from rest to 75 km/h in 5.0 s. What is the magnitude of its average acceleration?

If the velocity of an object is zero, does it mean that the acceleration is zero? If the acceleration is zero, does it mean that the velocity is zero?

The 3 Kinematic equations There are 3 major kinematic equations than can be used to describe the motion in DETAIL. All are used when the acceleration is CONSTANT.

Kinematic #1

a) How fast is the boat moving after accelerating for 5 seconds? Kinematic #1 Example: A boat moves slowly out of a marina (so as to not leave a wake) with a speed of 1.50 m/s. As soon as it passes the breakwater, leaving the marina, it throttles up and accelerates at 2.40 m/s/s. a) How fast is the boat moving after accelerating for 5 seconds? What do I know? What do I want? vo= 1.50 m/s v = ? a = 2.40 m/s/s t = 5 s 13.5 m/s

b) How far did the boat travel during that time? Kinematic #2 b) How far did the boat travel during that time? 37.5 m

Does all this make sense? 13.5 m/s 1.5 m/s Total displacement = 7.50 + 30 = 37.5 m = Total AREA under the line.

Kinematic #3 Example: You are driving through town at 12 m/s when suddenly a ball rolls out in front of your car. You apply the brakes and begin decelerating at 3.5 m/s/s. How far do you travel before coming to a complete stop? What do I know? What do I want? vo= 12 m/s x = ? a = -3.5 m/s/s V = 0 m/s 20.57 m

Which variable is NOT given and Examples How long does it take a car at rest to cross a 35.0 m intersection after the light turns green, if the acceleration of the car is a constant 2.00 m/s/s? Which variable is NOT given and NOT asked for? Final Velocity What do I know? What do I want? vo= 0 m/s t = ? x = 35 m a = 2.00 m/s/s 5.92 s

Which variable is NOT given and Examples A pitcher throws a fastball with a velocity of 43.5 m/s. It is determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body when the ball is at rest to the point of release. Calculate the acceleration during his throwing motion. Which variable is NOT given and NOT asked for? TIME What do I know? What do I want? vo= 0 m/s a = ? x = 2.5 m v = 43.5 m/s 378.5 m/s/s

Falling Objects Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity. This is one of the most common examples of motion with constant acceleration.

Falling Objects In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

Falling Objects The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2.

Figure 2-22 Tossing an object up An object thrown into the air leaves the thrower's hand at A, reaches its maximum height at B, and returns to the original position at C. Examples 2-12, 2–13, 2–14, and 2–15.

Kinematics for the VERTICAL Direction All 3 kinematics can be used to analyze one dimensional motion in either the X direction OR the y direction.

Which variable is NOT given and Examples A stone is dropped at rest from the top of a cliff. It is observed to hit the ground 5.78 s later. How high is the cliff? Which variable is NOT given and NOT asked for? What do I know? What do I want? -163.7 m H =163.7m

Graphs of (a) y vs. t (b) v vs. t for a ball thrown upward.

Graphical Analysis of Linear Motion This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.

Graphical Analysis of Linear Motion On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point. What would an a vs. t graph look like?

The velocity of an automobile as a function of time, starting from a dead stop. The jumps in the curve represent gear shifts.

Graphical Analysis of Linear Motion The displacement, x, is the area beneath the v vs. t curve.

The shaded area represents the displacement during the time interval t = 2.0s to t = 6.0s.