One-Dimensional Motion

Motion Graphs

Describing the motion of an object is occasionally hard to do with words.
Sometimes graphs help make motion easier to picture, and therefore understand.

Remember:

  • Motion is a change in position measured by distance and time.
  • Speed tells us the rate at which an object moves.
  • Velocity tells the speed and direction of a moving object.
  • Acceleration tells us the rate speed or direction changes.

    •  

    Distance-Time Graphs

    Plotting distance against time can tell you a lot about motion. Let's look at the axes:

    Time is always plotted on the X-axis (bottom of the graph).
    The further to the right on the axis, the longer the time from the start.

    Distance is plotted on the Y-axis (side of the graph).
    The higher up the graph, the further from the start.

     

    If an object is not moving, a horizontal line is shown on a distance-time graph.

    Time is increasing to the right, but the distance does not change. It is not moving.
    We say it is At Rest.

    If an object is moving at a constant speed, it means it has the same increase in distance in
    a given time- it will be a straight line. The slope of the line will indicate how quickly the object moved.

     

    Time is increasing to the right, and distance is increasing constantly with time.
    The object moves at a constant speed.

    Constant speed is shown by straight lines on a graph.

    Let’s look at three moving objects: All of the lines in the graph show that each object
    moved the same distance, but the steeper dashed line got there before the others:

    A steeper line indicates a larger distance moved in a given time.
    In other words, higher speed.

    All lines are straight, so both speeds are constant.

     

    Graphs that show acceleration look different from those that show constant speed.

    The line on this graph is curving upwards. This shows an increase in speed,
    since the line is getting steeper:

    In other words, in a given time, the distance the object moves is changing
    (getting larger). It is accelerating.

     

    Graphs Summary:

    A distance-time graph tells us how far an object has moved with time.

    • The steeper the graph, the faster the motion.
    • A horizontal line means the object is not changing its position - it is not moving,
      it is at rest.
    • A downward sloping line means the object is returning to the start.

     

    A great online review game of motion graphing.
    Motion graphing review:
    http://www.theuniverseandmore.com/

    Interpreting Graphs

    • Each point on the graph shows the object’s location.
    • Since v= d/t, slope of this graph is the velocity
    • Average velocity is the slope between the two points being averaged
    • Instantaneous speed of a point is the slope of that section on line
    • A flat line is at rest
    • Sloping upwards is a positive velocity (going forward)
    • Sloping down is a negative velocity (going in reverse)
    • An up-curving line is increasing its velocity (accelerating)
    • A down-curving line is decreasing its velocity (accelerating)
    • Total distance traveled is the sum of all the ups and downs
    • Total displacement is adding all the (+) and (-) ups and downs.

    • Flat lines represent a constant, steady speed
    • Lines sloping up are positive acceleration- going faster
    • Lines sloping down are decelerating (slowing down)
    • The slope of these lines are the acceleration.
    • Distance traveled is the area under the graph or section of the curve.

    • Acceleration is a measure of how hard you press on the gas pedal
    • Velocity is the area under the graph
    • This graph shows that you are pressing the pedal harder and harder.

    Displacement Graphs worksheet

    Interpreting Graphs notes sheet

    Graphing Morion worksheets 1 & 2

    Note: assume all examples, discussions and problems are
    without friction and air resistance. There are equations that
    take these forms of friction into account, but in order to learn
    the basics, we must start simple and under ideal circumstances.

    Acceleration due to Gravity Lab

    • The Spark Timer uses a small spark to burn a mark onto paper tape 60 times a second.
    • MAKE SURE TIMERS ARE SET TO 60Hz
    • Thread a length of spark tape through the timer and then tape it onto the mass.
    • When the weight falls, it will pull the tape through the timer leaving a trail
      of dots at measured intervals.
    • Write down the object that you drop.
    • Circle the first dot that is clearly separated- this will be our zero starting point.
    • Circle every 3rd dot
    • At 1/60th second, every 3rd = .05 sec
    • Set up the data table
    t
    d
    0.00
    0.05
    0.10
    0.15
    0.20
    ...
    • Measure the distance from the “zero” dot and enter it into a data table.
    • Make a d vs t graph
      • Use the entire graph paper
      • Work in pencil
      • Connect dots with a smooth curve
      • Every person makes their own
    • To calculate speed:
      • We want the instantaneous speed at the moment that the dot was made
        rather than the average speed since the beginning.
      • Measure the distance between the dots before and after your circled dot.
      • This value will be called Δd
      • This will give you the distance the mass dropped during the 2/60ths of
        a second surrounding the dot.
      • v = Δd ÷ 2/60sec. v is also more easily calculated as Δd x 30/sec
    • Fill in two more columns and graph
    t
    d
    Δd
    v
    0.00
    		   
    0.05
    		   
    0.10
    		   
    0.15
    		   
    0.20
    		   
    ...
    		   
    • Questions to consider in your conclusion:
      • How did distance traveled, Δd and velocity change through the experiment?
      • How did your measured acceleration compare with those of your classmates.
      • What was the correlation between mass and acceleration?
      • Predict how the results would change if you doubled the weight.
      • Be sure all measurements and calculations have units

    Astronaut dropping a feather and a hammer on the moon:

    Average Velocity Teaser:

    Suppose in making a round trip you travel at uniform speed of 30 mph from A to B
    which are separated by 100 miles, and return from B to A at a uniform rate of 60 mph.
    What would be your average speed for the round trip? (Careful!)

    “The Arsenal”

    • v = d/t: you already know
    • a = Δv/t
      Acceleration is a change in velocity
      Speeding up
      Slowing down: AKA decelerating or a “negative acceleration”
      Changing direction (more on that MUCH later)
      Units are m/sec2
      a = Δv/t problem:

    A car goes from 0 m/s to 160 m/s in 5 seconds
    What is the acceleration?

    a = (160m/s - 0m/s)/5sec
    a = 32m/s2

    Note on acceleration: It seems strange to have seconds "squared" since it is an
    intangible quantity. If it helps, 32m/s2 can also be read as
    32 meters per second per second, which means “for every second
    the velocity changes by 32m/s”.

    Arsenal 1 worksheet: v = d/t

    Arsenal 2 worksheet: a = Δv/t

    vf = vi + at

    It is usually v = at but the vi is added in incase you are not starting from a rest.
    “v is where it’s at”

    vf = vi + at problem: (actually starting at rest so it is vf = at)
    What is the exit velocity of a bullet after it travels down a rifle barrel with
    an acceleration of 2.0 x 105 m/s2 in a time of .003 seconds?

    vf = vi + at problem:
    A rocket car is cruising down a road at 40 m/s using only the conventional
    engine in the car. The driver then engages the rockets and the car accelerates
    at a rate of 10 m/s2 until the engines cut off after 10 seconds of thrust.
    What is the final velocity?
    (Professional driver on a closed course. Do not attempt)

    Arsenal 3 worksheet: v = at

    d = vit + ½at2

    It is usually d = ½ at2 but you add in the vit in case you are already moving.

    d = vit + ½at2 problem: (actually starting at rest so it is d = ½at2)
    A model rocket provides an acceleration of 35 m/s2 for 6 seconds.
    How high does the rocket reach when the engine cuts off?

    d = vit + ½at2 problem:
    A Klingon Bird of Prey is hurtling through space at 250 km/sec when it is
    caught in the gravitational pull of the sun. At its current position, the
    Bird of Prey’s acceleration due to the Sun’s gravity is 200 m/s2.
    The ship is subjected to this acceleration for 90 seconds before it slams into
    the surface of the sun with an unimpressive “pop”. What was its final velocity?

    Arsenal 4 worksheet: d = vit + ½at2 where vi = 0

    Arsenal 5 Worksheet: d = vit + ½at2

    vf2 = vi2 +2ad

    Again, this equation is usually vf2 = 2ad if you are starting from rest.

    vf2 = vi2 +2ad problem: (actually starting at rest so it is vf2 = 2ad)
    A potato cannon accelerates a spud down a 1.2m tube at a rate of 1041m/s2.
    What is its final muzzle velocity?
    Don’t forget the square root.

    Arsenal 6 worksheet: vf2 = vi2 +2ad where vi = 0

    Arsenal 7 worksheet: vf2 = vi2 +2ad

    Example vf2 = vi2 +2ad problem where vi2 is not zero


    Photo: Phil Medina, Jones Beach Memorial Day Air Show 2011
    A10 Thunderbolt "Warthog" piloted by Major Thorpe.

    An A10 Thunderbolt Warthog dives down for a strafing run on an enemy target
    at a velocity of 89 m/s. It fires its coke-bottle-sized bullets down its 3 meter cannon
    with an acceleration of 121,000 m/s2. What is the bullet’s velocity when it slams into its target?

     

    The Arsenal: These are your weapons to
    deal with motion problems.

    v = d/t
    a = Δv/t
    vf = vi + at
    d = vit + ½at2
    vf2 = vi2 +2ad

    Arsenal 8 worksheet: "Weapons of Math Destruction" Multiple equation problems

    Tips for Solving a Math problem

    • Make a list of all known variables
    • Identify what needs to be found
    • Select an equation that will solve the problem
    • Cross out whatever doesn’t apply (ex.: starting at rest)
    • Rearrange your equation to solve for the unknown (algebra)
    • Include units
    • Check algebra by analyzing units
    • Crunch the numbers
    • Do not leave answers as fractions or radicals
    • Circle your answer

    Don’t mix up the v’s

    Each v means something different:

    • [Average velocity]- ‘nuff said. This will not help you if you need to find the
      beginning or ending velocities.
    • vi: initial velocity- velocity at the beginning of the movement.
    • vf: final velocity- velocity at the end of movement.
    • Δv: change in velocity- the differenc ebetween the initial and final velocities or vf - vi

    Free Fall

    Free fall problems can often be solved by using:
    vf = vi + at     to find v or t
    d = vit + ½at2     to find d or t
    or
    vf2 = vi2 +2ad to find v or d

    However, the vi portion of the equation will turn into zero and can be crossed
    out, if it drops from rest (vi = 0). Mathematically, it will also work out the same
    way (crossing out the vi portion) if the object is thrown upwards and reaches the
    peak (where its velocity will = 0)

    Free Fall Problem:

    You have been sent to Pluto by NASA to fix a 90-meter high radio antenna.
    The antenna must function in order for NASA to communicate with astronauts
    sent to explore the planet Xena and its moon Gabrielle. While working on the
    antenna, you drop your wrench. It takes the wrench 21.2 seconds to fall to the
    cold surface of the Dwarf Planet. Calculate the acceleration due to gravity on Pluto.

     

    If you jump up at 5.4 m/s, how long will you stay in the air (round trip)?

     

    Could you jump back to the top of the antenna if you jump at 5.4 m/s?

    Free Fall Problems 1
    Free Fall Problems 2
    Free Fall Problems 3

    Will it speed up or slow down?

    Establish which direction represents + or -

    6 5 4 3 2 1 0 1 2 3 4 5 6
    -<------------------------>+

    v+ a+
    v- a-
    (same sign)
    		 = faster speed
    v+ a-
    v+ a-
    (opposite sign)
    		 = slower speed


     

    Over fifteen years of
    Medina On-Line
    1994-2011