Calculating Acceleration from a Velocity vs. Time Graph

Velocity vs. Time Graph Fundamentals

  • Graph Axes and Variables:
    • Vertical Axis (Y-axis): Velocity, measured in meters per second (m/s\text{m/s}).
    • Horizontal Axis (X-axis): Time, measured in seconds (sec\text{sec}), ranging from 0 sec0\,\text{sec} to 20 sec20\,\text{sec}.
  • Physical Significance of Graph Slope:
    • The slope of any segment on a Velocity vs. Time graph represents the acceleration of the object during that time interval.
    • Acceleration is expressed in units of meters per second squared (m/s2\text{m/s}^2).
  • Mathematical Formula for Slope:
    • Slope=ΔvΔt\text{Slope} = \frac{\Delta v}{\Delta t}
    • Slope=v2−v1t2−t1\text{Slope} = \frac{v_2 - v_1}{t_2 - t_1}
    • Where Δv\Delta v is the change in velocity (m/s\text{m/s}) and Δt\Delta t is the change in time (sec\text{sec}).

Step-by-Step Graph Slope Calculations

  • Instructor Demonstration ("I DO"): Slope at 3 seconds3\,\text{seconds}:

    • Problem: Find the slope of the line at 3 seconds3\,\text{seconds}.
    • Calculated Slope: −1 m/s2-1\,\text{m/s}^2

  • Guided Practice ("WE DO"): Slope at 17 seconds17\,\text{seconds}:

    • Problem: Find the slope of the line at 17 seconds17\,\text{seconds}.
    • Calculated Slope: 3 m/s23\,\text{m/s}^2

  • Independent Practice ("YOU DO"): Slope at 11 seconds11\,\text{seconds}:

    • Problem: Find the slope of the line at 11 seconds11\,\text{seconds}.
    • Time Difference (Δt\Delta t):
      • 12−11=1 s12 - 11 = 1\,\text{s}
    • Velocity Difference (Δv\Delta v):
      • −6−(−4)=−6+4=−2 m/s-6 - (-4) = -6 + 4 = -2\,\text{m/s}
    • Slope Calculation:
      • Slope=−2 m/s1 s=−2 m/s2\text{Slope} = \frac{-2\,\text{m/s}}{1\,\text{s}} = -2\,\text{m/s}^2
    • Result: −2 m/s2-2\,\text{m/s}^2

  • Independent Practice ("YOU DO"): Slope at 14 seconds14\,\text{seconds}:

    • Problem: Find the slope of the line at 14 seconds14\,\text{seconds}.
    • Time Difference (Δt\Delta t):
      • 14−13=1 s14 - 13 = 1\,\text{s}
    • Velocity Difference (Δv\Delta v):
      • −5−(−8)=−5+8=3 m/s-5 - (-8) = -5 + 8 = 3\,\text{m/s}
    • Slope Calculation:
      • Slope=3 m/s1 s=3 m/s2\text{Slope} = \frac{3\,\text{m/s}}{1\,\text{s}} = 3\,\text{m/s}^2
    • Result: 3 m/s23\,\text{m/s}^2

  • Independent Practice ("YOU DO"): Slope at 19 seconds19\,\text{seconds}:

    • Problem: Find the slope of the line at 19 seconds19\,\text{seconds}.
    • Velocity Difference (Δv\Delta v):
      • 7−7=0 m/s7 - 7 = 0\,\text{m/s}
    • Time Difference (Δt\Delta t):
      • 20−19=1 s20 - 19 = 1\,\text{s}
    • Slope Calculation:
      • Slope=0 m/s1 s=0 m/s2\text{Slope} = \frac{0\,\text{m/s}}{1\,\text{s}} = 0\,\text{m/s}^2
    • Result: 0 m/s20\,\text{m/s}^2

  • Independent Practice ("YOU DO"): Slope at 12.5 seconds12.5\,\text{seconds}:

    • Problem: Find the slope of the line at 12.5 seconds12.5\,\text{seconds}.
    • Time Difference (Δt\Delta t):
      • 13−12=1 s13 - 12 = 1\,\text{s}
    • Velocity Difference (Δv\Delta v):
      • −8−(−7)=−8+7=−1 m/s-8 - (-7) = -8 + 7 = -1\,\text{m/s}
    • Slope Calculation:
      • Slope=−1 m/s1 s=−1 m/s2\text{Slope} = \frac{-1\,\text{m/s}}{1\,\text{s}} = -1\,\text{m/s}^2
    • Result: −1 m/s2-1\,\text{m/s}^2