Motion in One Direction Chapter 2

2-1: Displacement and Velocity Main Objectives:  Describe motion in terms of frame of reference, displacement, time, and velocity.  Calculate the displacement of an object traveling at a known velocity for a specific time interval.  Construct and interpret graphs of position versus time.

Chapter 2 Introduction The motion of objects is an important part of everyday life  Motion was the first aspect of the physical world to be thoroughly studied Our modern understanding of motion was first established during the sixteenth and seventeenth centuries  Galileo Galilei ( ) and Isaac Newton ( ) provided significant contributions to this understanding

Chapter 2 Introduction The study of the motion of objects, force, and energy form the field called mechanics  kinematics – the description of how objects move  dynamics – studies why objects move as they do

Speed average speed – the distance traveled divided by the time it takes to travel this distance average speed = Example: A car that travels 240km in 3h has an average speed of 80km/h. distance traveled time elapsed

Converting Units It is often necessary to convert between units  For example converting km/h to m/s This can be accomplished by using the technique of dimensional analysis which was introduced in chapter one

Converting Units When using dimensional analysis, one must make sure to: 1. Use correct conversion factors (must be equal to one), and 2. Set-up the problem to cancel out the appropriate units

Converting Units Example: Convert 80km/h into m/s. Practice: Convert 35.0m/s into km/h. 80km x 1m x 1h x 1min = 22m/s 1h km 60min 60s

Frame of Reference Every measurement must be made with respect to a frame of reference – this provides the information necessary to understand the measurement  The frame of reference is a way of comparing measurements Example: You are on a train that is traveling at a speed of 80km/h. A passenger on the train walks by you at a speed of 5km/h. Does the passenger have an average speed of 5km/h or 85km/h?

Frame of Reference Both average speeds of the passenger are correct if given the proper frame of reference.  The person has a speed of 5km/h relative to the train.  The person has a speed of 85km/h relative to the ground. The frame of reference should be specified whenever there might be confusion.

Frame of Reference Other physical quantities are dependent on a frame of reference.  Distance: Columbus, OH is 230km away from Cleveland  Direction: The plane was flying at 1000km/h heading northeast from Cleveland.

Frame of Reference In physics, a set of coordinate axes are used to represent a frame of reference.

Vectors and Scalars vector – a quantity which has a direction as well as a magnitude  Examples: velocity, displacement, force, momentum scalar – a quantity that only has a magnitude  Example: time, temperature, energy

Average Velocity and Displacement The terms speed and velocity are often used interchangeably, but in physics there is a distinction  velocity – used to signify both the magnitude (the numerical value) of how fast an object is moving and the direction the object is moving Speed is just a magnitude

Average Velocity and Displacement In addition, average velocity is defined in terms of displacement rather than total distance traveled average velocity = displacement time elapsed

Average Velocity and Displacement displacement (∆x or ∆y) – an object’s change in position displacement = final position – initial position ∆x = x – x 0 or ∆y = y – y 0 Example: A person walks 70m to the east, turns around, and walks back (west) 30m.  The total distance traveled is 100m, but the displacement is only 40m because the person is 40m away from the starting point

Average Velocity and Displacement Example (cont.): Assume the walk in the following example took 80s.  average speed = 100m/80s = 1.3m/s  average velocity (magnitude) = 40m/80s = 0.50 m/s This discrepancy between speed and the magnitude of velocity only occurs when dealing with average values

Average Velocity and Displacement When an object moves from on point to another:  Displacement = x – x 0 = ∆x  Elapsed Time = t – t 0 = ∆t Therefore: v = =  The sign for displacement (+/-) can indicate the direction of motion ∆x x – x 0 ∆t t – t 0 _

Average Velocity and Displacement Example: The position of a runner as a function of time is plotted as moving along the x-axis of a coordinate system. During a 3.00s interval, the runner’s position changes from x 0 = 50.0m to x = 30.5m. What was the runner’s average velocity?

Instantaneous Velocity instantaneous velocity – the velocity at any instant of time  Example: If you travel along a straight road for 150 km in 2.0 h, the magnitude of your average velocity would be 75 km/h. However, it is unlikely that your velocity was 75km/h at every instant.

Instantaneous Velocity An object with a uniform (constant) velocity over a particular time would have the same instantaneous velocity at any instant as its average velocity.

Position vs. Time Graphs On a position-time graph, an object with a uniform velocity would for a straight line.  The velocity (constant) would be equal to the slope of the line.

Position vs. Time Graphs An object without a uniform velocity would not form a straight line on a position-time graph  The instantaneous velocity of the object would be equal to the slope of a tangent line at any point along the curve

Position vs. Time Graphs

Acceleration An object whose velocity is changing is said to be accelerating.  This can be a change in the magnitude of velocity Example: A car’s velocity increasing from 0 to 80km/h  Or it can be a change in the object’s direction A plane traveling at a constant 600km/h turns from north to northeast.

Acceleration average acceleration – the rate of change of velocity  It is the change in velocity divided by the time taken to make this change average acceleration = a = = change of velocity time elapsed _ v – v 0 ∆v t – t 0 ∆t

Acceleration (Units) Usually the same unit of time is used when expressing acceleration (the same unit of time for the velocity and for the ∆t). = =  Complete the previous example using m/s for the velocity units. m/s m m s s s s 2

Acceleration Example: A car accelerates along a straight road from rest to 60km/h in 5.0s. What is the magnitude of its average acceleration?

Acceleration Example: A car is moving along a straight highway (which can be represented as right along an x- axis) and the driver puts on the brakes. If the initial velocity is v 0 = 15.0 m/s and it takes 5.0 s to slow down to v = 5.0 m/s, what was the car’s average acceleration in m/s 2 ?

Acceleration Acceleration is a vector quantity – it has a magnitude and a direction

Uniform (Constant) Acceleration Objects are said to have uniform acceleration when the magnitude of the acceleration is constant and the motion is in a straight line  The relationships between position, velocity, constant acceleration, and the time in motion can be expressed with various equations.

Uniform Acceleration Variables: x 0 = initial positionx = position at time, t v 0 = initial velocityv = velocity at time, t t = time elapsed

Uniform Acceleration - Equations The average velocity of an object with uniform acceleration will be halfway between the initial and final velocities. Equation 1:v = v 0 + v 2 _

Uniform Acceleration - Equations The final velocity of an object can be calculated when its acceleration, initial velocity, and the time elapsed is known. Equation 2: v = v 0 + at

Uniform Acceleration - Equations The position of an object after a certain amount of time can be calculated when it is undergoing constant acceleration if you know the initial velocity. Equation 3: x = x 0 + v 0 t + ½ at 2

Uniform Acceleration - Equations The final velocity can be calculated without knowing the time elapsed if the initial velocity, displacement, and acceleration are known. Equation 4: v 2 = v a(x - x 0 ) *Since a square root can be either + or -, you must determine the direction of motion by comparing the directions of the initial velocity and acceleration

Falling Bodies A common example of uniformly accelerated motion is an object in free fall near the Earth’s surface  free fall – the motion of a body when only the force due to gravity is acting on the body Until Galileo’s time, it was believed that heavier objects fall faster than lighter objects and that the speed of fall is proportional to how heavy the object is.

Falling Bodies Why is it that a feather will take longer to hit the ground than an apple when dropped from the same height?

Falling Bodies Galileo’s Hypothesis: at a given location on the Earth and in the absence of air resistance, all objects fall with the same uniform acceleration  This acceleration is called the acceleration due to gravity on the Earth It is approximately: g = 9.80 m/s 2, downward or g = m/s 2

Falling Bodies In the absence of air resistance, all objects fall with the same constant acceleration regardless of their masses.

Falling Bodies The effects of air resistance are often small and will be disregarded in most examples.  However, the effects of air resistance are noticeable even on reasonably heavy objects if the distance the object falls is very large

Falling Bodies If an object falls far enough in air, it will reach a maximum velocity called terminal velocity.  Terminal velocity is reached when the force of air resistance (which increases with speed) reaches a magnitude equal to the force of gravity.

Falling Bodies Free fall acceleration is constant during upward and downward motion.

Falling Bodies

You can use the equations for objects undergoing uniformly accelerated motion to solve problems involving free falling objects  In these situations, “a” will be equal to g = m/s 2 near the Earth’s surface Also, since the motion is vertical, x will be replaced with y and x 0 is replaced with y 0.

Example 1: Suppose that a ball is dropped from a tower 70.0 m high. How far will it have fallen after 1.00, 2.00, and 3.00 s? Assume y is negative with downward motion.

Example 2: Suppose the ball in the previous example was thrown downward with a speed of m/s instead of being dropped. (a) What would be the ball’s position after 1.00 s and after 2.00 s? (b) What would its speed after 1.00 s and 2.00 s. Compare these speeds to the speeds of a ball that was dropped (v 0 = 0).

Displacement & Velocity vs. Time Graphs