Graphing Motion Warm-Up Kinematics Study Guide

Graphing Motion Warm-Up Assessment Structure

• Worksheet Component Breakdown and Point Values:

  • Position vs. Time ($x-t$) Graph Section: Initial plotting grid and analysis.
  • Velocity vs. Time ($v-t$) Graph Section: Worth $20\,\text{points}$.
  • Acceleration vs. Time ($a-t$) Graph Section: Worth $20\,\text{points}$.
  • Conceptual and Quantitative Questions Section: Contains $12$ questions worth $5\,\text{points}$ each, totaling $60\,\text{points}$.
  • Total Assessment Point Value: $100\,\text{points}$.

• Position vs. Time ($x-t$) Axis Grid Specifications:

  • Time Axis ($t$): Represented on the horizontal axis in units of seconds ($s$). Axis labels span from $0$ to $18$ with tick marks at interval increments of $2\,s$ ($0\,s$, $2\,s$, $4\,s$, $6\,s$, $8\,s$, $10\,s$, $12\,s$, $14\,s$, $16\,s$, $18\,s$).
  • Position Axis ($x$): Represented on the vertical axis in units of meters ($m$). Axis labels span from $0$ to $10$ with tick marks at integer increments of $1\,m$ ($0\,m$, $1\,m$, $2\,m$, $3\,m$, $4\,m$, $5\,m$, $6\,m$, $7\,m$, $8\,m$, $9\,m$, $10\,m$).

Motion Analysis Questions

• Question 1:

  • Prompt: What was the initial velocity of the object?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Initial velocity ($v_0$ or $v_i$) is evaluated at time $t = 0\,s$. On a position vs. time graph, it corresponds to the slope of the position line at $t = 0\,s$, defined mathematically by $v_i = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$. On a velocity vs. time graph, it is the vertical intercept at $t = 0\,s$.

• Question 2:

  • Prompt: At what times was the object stopped?
  • Point Value: $5\,\text{points}$
  • Analytical Method: An object is at rest ($v = 0\,m/s$) when:
  • On a position vs. time ($x-t$) graph, the slope is zero (horizontal line segment, $\frac{\Delta x}{\Delta t} = 0\,m/s$).
  • On a velocity vs. time ($v-t$) graph, the curve crosses or lies directly on the horizontal time axis ($v(t) = 0\,m/s$).

• Question 3:

  • Prompt: At what times was the object moving left?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Motion to the left indicates a negative direction of motion, corresponding to negative velocity ($v < 0\,m/s$):
  • On an $x-t$ graph, this is indicated by a negative slope (position decreases as time advances).
  • On a $v-t$ graph, this is represented by regions where the velocity curve lies below the horizontal time axis ($v(t) < 0\,m/s$).

• Question 4:

  • Prompt: At what times was the object moving right?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Motion to the right indicates a positive direction of motion, corresponding to positive velocity ($v > 0\,m/s$):
  • On an $x-t$ graph, this is indicated by a positive slope (position increases as time advances).
  • On a $v-t$ graph, this is represented by regions where the velocity curve lies above the horizontal time axis ($v(t) > 0\,m/s$).

• Question 5:

  • Prompt: At what times was the object slowing down?
  • Point Value: $5\,\text{points}$
  • Analytical Method: An object is slowing down when its speed $|v|$ approaches zero ($0\,m/s$):
  • On an $x-t$ graph, the line flattens out toward a horizontal slope.
  • On a $v-t$ graph, the line approaches the zero axis, meaning velocity and acceleration vectors have opposite signs ($v \cdot a < 0$).

• Question 6:

  • Prompt: What was the acceleration from $2-3$ seconds?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Average acceleration over the time interval from $t = 2\,s$ to $t = 3\,s$ is calculated using $a = \frac{v(3\,s) - v(2\,s)}{3\,s - 2\,s} = \frac{\Delta v}{\Delta t}$. On a velocity vs. time graph, this is equal to the slope of the velocity curve between $t = 2\,s$ and $t = 3\,s$.

• Question 7:

  • Prompt: What was the displacement from $2-3$ seconds?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Displacement ($\Delta x$) over the interval from $t = 2\,s$ to $t = 3\,s$ is given by:
  • From an $x-t$ graph: $\Delta x = x(3\,s) - x(2\,s)$.
  • From a $v-t$ graph: the area under the velocity curve bounded between $t = 2\,s$ and $t = 3\,s$, defined as $\Delta x = \int_{2\,s}^{3\,s} v(t)\,dt$.

• Question 8:

  • Prompt: What was the average velocity for the first three seconds?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Average velocity ($v_{avg}$) from $t = 0\,s$ to $t = 3\,s$ is total displacement divided by elapsed time: $v_{avg} = \frac{x(3\,s) - x(0\,s)}{3\,s - 0\,s} = \frac{\Delta x}{\Delta t}$.

• Question 9:

  • Prompt: At what times was the velocity of the object constant?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Constant velocity implies zero acceleration ($a = 0\,m/s^2$):
  • On an $x-t$ graph, constant velocity appears as straight diagonal line segments with a constant slope.
  • On a $v-t$ graph, constant velocity appears as flat, horizontal line segments (zero slope, $\frac{\Delta v}{\Delta t} = 0\,m/s^2$).

• Question 10:

  • Prompt: What was the total displacement of the object?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Total displacement ($\Delta x_{total}$) is the net change in position from initial time $t_i$ to final time $t_f$: $\Delta x_{total} = x(t_f) - x(t_i)$. On a velocity vs. time graph, it is calculated as the net area between the velocity curve and the time axis: $\Delta x_{total} = \int_{t_i}^{t_f} v(t)\,dt$

• Question 11:

  • Prompt: What was the total distance travelled?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Total distance ($d$) measures the total length of path traveled without regard to direction: $d = \sum |\Delta x_i|$. On a velocity vs. time graph, it is evaluated as the total absolute area bounded by the curve: $d = \int_{t_i}^{t_f} |v(t)|\,dt$

• Question 12:

  • Prompt: What was the change in velocity from $0-10$ seconds?
  • Point Value: $5\,\text{points}$
  • Analytical Method: Change in velocity ($\Delta v$) from $t = 0\,s$ to $t = 10\,s$ is calculated as $\Delta v = v(10\,s) - v(0\,s)$. On an acceleration vs. time ($a-t$) graph, it equals the definite integral or area under the acceleration curve: $\Delta v = \int_{0\,s}^{10\,s} a(t)\,dt$

Core Kinematic Relationships and Graph Rules

• Position vs. Time ($x-t$) Graph Properties:

  • The vertical axis value represents the position ($x$) in meters ($m$).
  • The slope of the line equals instantaneous velocity: $v(t) = \frac{dx}{dt}$.
  • Positive slope ($\frac{\Delta x}{\Delta t} > 0$) signifies movement in the positive direction (rightward motion).
  • Negative slope ($\frac{\Delta x}{\Delta t} < 0$) signifies movement in the negative direction (leftward motion).
  • Zero slope ($\frac{\Delta x}{\Delta t} = 0$) signifies zero velocity (object stopped).
  • Concavity reflects acceleration: concave up ($\frac{d^2 x}{dt^2} > 0$) means positive acceleration; concave down ($\frac{d^2 x}{dt^2} < 0$) means negative acceleration.

• Velocity vs. Time ($v-t$) Graph Properties:

  • The vertical axis value represents instantaneous velocity ($v$) in meters per second ($m/s$).
  • The slope of the line equals instantaneous acceleration: $a(t) = \frac{dv}{dt}$.
  • Area under the line equals displacement: $\Delta x = \int v(t)\,dt$.
  • Position above time axis ($v(t) > 0\,m/s$) represents motion to the right.
  • Position below time axis ($v(t) < 0\,m/s$) represents motion to the left.
  • Points on the horizontal axis ($v(t) = 0\,m/s$) represent moments when the object is stopped.
  • Speeding up occurs when the velocity graph moves further away from the horizontal axis ($|v|$ increases).
  • Slowing down occurs when the velocity graph moves closer to the horizontal axis ($|v|$ decreases).

• Acceleration vs. Time ($a-t$) Graph Properties:

  • The vertical axis value represents instantaneous acceleration ($a$) in meters per second squared ($m/s^2$).
  • Area under the curve equals the net change in velocity: \Delta v = \int a(t)\,dt$.\n * Horizontal line at a = 0\,m/s^2 indicates motion at constant velocity or complete rest.\n\n* **Fundamental Mathematical Kinematic Formulas**:\n * Instantaneous Velocity: v(t) = \frac{dx}{dt}\n * Instantaneous Acceleration: a(t) = \frac{dv}{dt}\n * Average Velocity: v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}\n * Average Acceleration: a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}\n * Displacement from Velocity Integral: \Delta x = \int_{t_i}^{t_f} v(t)\,dt\n * Total Distance from Speed Integral: d = \int_{t_i}^{t_f} |v(t)|\,dt\n * Velocity Change from Acceleration Integral: \Delta v = \int_{t_i}^{t_f} a(t)\,dt$$