Chapter 2 Motion in One Dimension

Key Objectives  Define Motion in One Dimension  Differentiate Distance v Displacement  Compare Velocity v Speed  Calculate Velocity and Acceleration  Interpret Graphs of Position v Time and Velocity v Time  Kinematic Equations (Constant Acceleration)  Free Fall & Gravity

Motion in One Dimension  One-dimesional motion is the simplest form of motion.  This is motion that takes place in two directions.  In physics, that is best described as going forwards or backwards only.

Frame of Reference Motion takes place over time. We must account for that time whenever we talk of any characteristic of motion. The motion of an object can be very difficult to describe because of all of its characteristics. Therefore we must establish a frame of reference, or system for specifying the precise location of an object. This is done by defining initial and final conditions for all motion. Initial x i = 0 units Final x f = x units

Distance v Displacement  Distance is defined as the total amount that an object traveled.  Distance does not have a direction.  For instance if you were to run around the track that circles the football field 4 times, your distance would be 1 mile.  Displacement is defined as a change in position.  Direction is included in displacement. This is done by setting your frame of reference.  Since displacement involves direction, it can be negative or positive.

Formula for Displacement Displacement is found by subtracting the initial position from the final position of the object. Remember that distance does not matter, it is simply final position and initial position. Δx = xfxf -xixi That is the Greek letter delta, which means “change” The variable x is used to describe horizontal motion and y is used for vertical motion. So even though you have gone around the track 4 times, your displacement will be … 0

Velocity  Speed is defined as the distance an object travels divided by the time interval of the motion.  Since distance has no direction, neither does speed.  An example of speed is 35 mph.  Velocity is defined as the displacement of an object divided by the total time period during which the displacement occurred.  Velocity does have a direction.  Therefore, velocity can be negative or positive depending on the displacement.  Examples of velocity are -35 mph or 35 mph left.

Formula for Velocity Remember that velocity is the displacement of an object divided by the total time it took for the object to achieve that displacement. v = Δx Δt = x f - x i t f - t i Because we are calculating the object's motion at two different times, this is an average velocity of the object. av g Instantaneous velocity is found at one instant in time and is only found using the power of calculus! Therefore instantaneous velocity can be larger, smaller, or the same magnitude as the average velocity.

Acceleration  Acceleration measures the rate of change in velocity.  That is acceleration can be defined as the amount the velocity changes divided by a given time interval.  Since displacement and velocity include direction, so does acceleration.  Again meaning that acceleration can be negative or positive, depending on whether the velocity increased or decreased in the direction that it was traveling. a = Δv Δt = v f - v i t f - t i av g

Position v Time The x-axis represents time in seconds, and the y-axis represents position in meters. Therefore, if you calculate the slope of the line y 2 – y 1 x 2 – x 1 actually takes displacement time which we call

Velocity v Time Again take the slope of this line y 2 – y 1 x 2 – x 1 actually takes velocity time which we call Initial Velocity

Area Under a Velocity-Time Graph For this type of a graph only, we can find the area under the curve to tell us another characteristic of motion. Area for a rectangle is: Area for a triangle is: A = bh A = ½ bh Regardless, the measurement of the base is:timex And the height is: velocity s x m/s Which leaves us with: m So area under this curve tells us displacement.

Kinematic Equations  That is just a physics term for motion with constant acceleration.  So to use the following equations, you must either know or assume constant acceleration of an object.  Any time the acceleration changes, you need to use a different equations.  Since acceleration is constant, that means that velocity increases at the same rate for each time interval and the displacement increases at the same rate for the same time interval.  These equations can be derived from a Velocity-Time Graph!

The Kinematics Equations ΔtΔta v i + v f = v f 2 = v i 2 + 2aΔx Δt (v f + v i ) 2 Δx = = v i Δt+ ½aΔt 2 Displacement as a function of velocity Velocity as a function of time Displacement as a function of time Velocity as a function of displacement

Gravity Free fall is defined as an object traveling through the air with no outside force acting on the object other than gravity. Gravity is a constant acceleration, no matter the mass of an object. That is as long as air resistance is ignored or not present. Because those provide an outside force! We assign a variable to the acceleration due to gravity so we can use it in our kinematic equations. g = m/s 2 Shows direction…down!!