Anonymous asked in Science & MathematicsPhysics · 3 months ago

Help with applying Newton's second law...?

So the scenario is:

A ball is thrown vertically upwards with initial speed u, and travels upwards under the influence of gravity and air resistance. Use the quadratic model of air resistance with the ball modelled as a sphere of effective diameter D.

We're asked to obtain the equation:

dv\dt = -k(v^(2) + (g\k)) but here's how far I get... (the T's are all supposed to be small ones)

Attachment image

1 Answer

  • NCS
    Lv 7
    3 months ago
    Favorite Answer

    I don't understand your question. The equation follows directly from Newton's Second:

    a = drag acceleration - gravitational acceleration

    dv/dt = kv² - g = k*(v² - g/k) → for quadratic case

    which is a first-order nonlinear ordinary differential equation.

    At terminal velocity, dv/dt = 0 and so terminal v = √(g/k)

    where it is subsequently shown that k = c₂D² / m.

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