Motion in 1-D: Describing Motion to Velocity-Time Graphs

Describing Motion

  • We analyze motion in terms of displacement, velocity, and acceleration.
  • A motion diagram is a series of images of a moving object that records its position after equal time intervals.
  • Four concepts in the study of motion: at rest, speeding up, slowing down, and constant speed.
  • Visual cue: identifying which diagram corresponds to each concept.

Particle Model and Coordinate Systems

  • Particle model: Replacing an object by a single point to describe its motion.
  • A coordinate system tells you where the zero point of the variable you are studying is located and the direction in which the values increase.
  • The origin is the point where the variables have the value zero.
  • Typically:
    • The horizontal direction is the x-axis.
    • The vertical direction is the y-axis.

Vectors and Scalars

  • Position vector: The length of a position vector is proportional to the distance from the origin and it points from the origin to the location of the moving object at a particular time.
  • Scalar quantity: A quantity that only tells you the magnitude (e.g., 25° C, 125 grams, 15 seconds).
  • Vector quantity: A quantity that has both magnitude and direction (e.g., velocity v, acceleration a).
  • Notation: v represents velocity and a represents acceleration.

Time Intervals and Displacements

  • Displacement is the distance and direction between two positions (a vector).
  • Displacement can be written as \Delta d = d1 - d0.
  • The length (magnitude) of the displacement vector is the distance between the two positions.
  • Distance is a scalar quantity (it has only magnitude, no direction).
  • Time interval is the difference between two times, written as \Delta t = t1 - t0.

Velocity and Acceleration

  • Average velocity, v = \frac{\Delta d}{\Delta t} = \frac{d1 - d0}{t1 - t0}.
  • Velocity is a vector in the same direction as the displacement.
  • Instantaneous velocity is the speed and direction of an object at a particular instant in time.
  • The sign of the average velocity depends on the chosen coordinate system.

Acceleration

  • Acceleration: an object in motion whose velocity is changing is said to be accelerating.
  • Average acceleration: a = \frac{\Delta v}{\Delta t} = \frac{v1 - v0}{t1 - t0}.
  • The sign of the acceleration is determined by whether the object is speeding up or slowing down.

Problem-Solving Strategy for Velocity and Acceleration (Problem-Solving Approach)

1) Sketch the problem: sketch the situation, establish a coordinate system, build a pictorial model with symbols.
2) Write the equation.
3) Write the solution.
4) Box the answer and check if your answer is reasonable.

Example Problem (Do Not Solve)

  • Sketch the following problem: A driver, going at a constant speed of 25\ \text{m/s}, sees a child suddenly run into the road. It takes the driver 0.4\ \text{s} to hit the brakes. The car then slows at a rate of 8.5\ \text{m/s}^2. What is the total distance the car moves before it stops? (Do not solve)

Graphing Motion in 1-D: Position-Time Graphs

  • You can use a position-time (p-t) graph to determine where and when the object is.
  • You can graph the motion of two or more objects on the same p-t graph.

Interpreting a Position–Time (p-t) Graph

  • Motion away from the origin in a + direction has a + velocity; motion in the - direction has a - velocity; no change in position has a 0 velocity.
  • Uniform velocity means displacements occur during successive equal time intervals; the p-t graph is a straight line.
  • Interpret the following situations: car driving; four walkers A–D (conceptual interpretation of their p-t behavior).

Interpreting a p-t Graph: Slope and Average Velocity

  • On a p-t graph, the slope of a straight line equals the average velocity:
    \text{Slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta d}{\Delta t} = \frac{d1 - d0}{t1 - t0} = v.
  • Note: the average velocity is not simply d/t read directly from the p-t graph.
  • A negative slope indicates negative velocity (opposite to the chosen + direction).

Graphing Velocity in 1-D

  • When velocity is not constant:
    • The object speeds up or slows down, so the average velocity for each successive time interval changes.
    • Instantaneous velocity is not equal to the average velocity over a finite time interval.
    • The instantaneous velocity at time t is the slope of the tangent to the position-time curve at that time:
      v(t) = \text{slope of the tangent to the p-t curve at } t.

Velocity–Time (v-t) Graphs

  • Uniform motion (constant velocity) is represented by a horizontal line on a v-t graph.
  • When two v-t lines cross, the two objects have the same velocity at that moment (they do not necessarily meet in space).
  • v-t graphs provide no direct information about position, though you can deduce displacement from the area under the curve.
  • Interpretation cues: constant velocity (horizontal line) vs increasing velocity (sloped line).

Displacement from a Velocity–Time Graph

  • For constant velocity, \Delta d = v \Delta t.
  • The area under a v-t curve represents displacement:
    \Delta d = \iint v \, dt \quad \text{(area under the curve)}.
  • For a rectangle on a v-t graph, the area is v \Delta t.

Acceleration and Average Acceleration (Revisited)

  • Average acceleration: a = \frac{\Delta v}{\Delta t} = \frac{v1 - v0}{t1 - t0}.
  • This ratio corresponds to the slope of the v-t graph.
  • Recall: velocity is the slope of the p-t graph.