Drag, Free-Form vs Non-Free-Form Fall, and Terminal Velocity

Free-form vs Non-free form (drag-influenced) motion

  • Free-form fall (no air resistance):

    • Net force F_net equals weight W
    • W = m g
    • Acceleration a = F_net / m = g (direction: downward)
    • In this case, weight is the only force acting, so the object accelerates at g

  • Non-free form (with air resistance/drag):

    • Forces: weight W downward, drag D upward (opposes motion)
    • Net force F_net = W − D
    • Acceleration a = F_net / m = (W − D) / m
    • Drag increases with speed, so as speed increases, D grows and a decreases
    • The net force is the difference between two forces, weight and drag; it must be smaller than the weight itself (since D > 0)

  • Weight and drag in the context of the given numbers

    • Given: Weight W = 20 N, Drag D = 5 N
    • Net force F_net = W − D = 20 − 5 = 15 N
    • Mass m can be found from W = m g, so m = W / g
    • With g ≈ 9.8 m/s^2, m ≈ 20 / 9.8 ≈ 2.04 kg
    • Therefore, acceleration a = F_net / m ≈ 15 / 2.04 ≈ 7.35 m/s^2 (downward)

  • Formula recap (key relationships)

    • Net force in non-free form: Fextnet=W−DF_{ ext{net}} = W - D
    • Weight: W=mgW = m g
    • Free-form (no drag): F<em>extnet=W⇒a=F</em>extnetm=Wm=gF<em>{ ext{net}} = W \Rightarrow a = \frac{F</em>{ ext{net}}}{m} = \frac{W}{m} = g
    • Non-free form (with drag): a=Fextnetm=W−Dma = \frac{F_{ ext{net}}}{m} = \frac{W - D}{m}
    • Terminal velocity condition: drag balances weight, so D(vt)=W=mgD(v_t) = W = m g
    • If drag depends on velocity as D=cv2D = c v^2 (quadratic drag), then terminal velocity is vt=mgcv_t = \sqrt{\dfrac{m g}{c}}
    • If drag is linear in velocity, e.g., D=kvD = k v, then at terminal velocity kv<em>t=mg⇒v</em>t=mgkk v<em>t = m g \Rightarrow v</em>t = \dfrac{m g}{k}

  • Drag and velocity dynamics

    • Drag grows with speed, so as v increases, D increases and F_net = W − D decreases
    • Acceleration a = Fnet / m decreases as Fnet decreases
    • When D approaches W, F_net → 0 and a → 0; velocity approaches a constant: terminal velocity
    • The direction of acceleration during fall remains downward until terminal velocity is reached

  • Terminal velocity in practice: feather vs coin (mass effect)

    • With the same shape/drag characteristics, heavier object has larger terminal velocity because weight is larger, so it takes a larger drag force to balance it
    • Consequence: heavier object reaches a higher terminal speed, and (depending on drag law) may reach it later or sooner depending on how acceleration changes over time
    • In many simple drag models, a lighter object reaches its (lower) terminal velocity sooner; a heavier object takes longer to accelerate to a higher terminal velocity

  • Worked scenario: two objects of different weights in non-free fall

    • First conclusion: lighter object reaches terminal velocity first (has a lower v_t)
    • Heavier object: reaches terminal velocity later but at a higher v_t
    • Explanation: terminal velocity is set by the condition D(vt) = W = m g; heavier object has larger W, so vt must be larger to make D(v) equal to W

  • Practical questions and quick checks

    • Question: Who reaches terminal velocity first?

    • Answer: The lighter object reaches terminal velocity first, but its terminal velocity is lower; the heavier object reaches its terminal velocity later but with a higher terminal velocity

    • Question: An object is said to fall at a constant velocity of 1000 km/h when the forces balance (i.e., zero acceleration). How is the air resistance determined?

    • At constant velocity, a = 0, so F_net = 0

    • Therefore, Drag must balance Weight: D = W = m g

    • To find D at any given v, you would use the drag model D(v) and solve D(v) = m g for v

  • Notes on speed regimes and relationships

    • In free-form (no drag): acceleration is constant and equal to g, regardless of speed
    • In non-free form (with drag): acceleration decreases as speed increases, eventually reaching zero at terminal velocity
    • Terminal velocity depends on mass, cross-sectional area, drag coefficient, fluid density, and drag-law exponent (e.g., quadratic vs linear drag)

  • Connections to foundational principles

    • Newton's laws: F = m a governs both free and drag-influenced falls
    • Weight as the marshalling force: W = m g
    • Drag as a resistive force arising from fluid dynamics, increasing with velocity

  • Final takeaways

    • Free form: a = g; net force equals weight
    • Non-free form: a = (W − D)/m; drag reduces acceleration as speed grows
    • Terminal velocity occurs when D(v_t) = W
    • Heavier objects tend to have higher terminal velocities because a larger weight requires a larger drag to balance it; lighter objects reach smaller terminal velocities
    • If you know the drag law, you can compute vt; if D = c v^2, then v</em>t=mgcv</em>t = \sqrt{\dfrac{m g}{c}}