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Find velocity of car/tank as fxn of time.

Concept: Variable mass system

Question

Find velocity of car/tank as fxn of time.
F=dpdtF = \frac{dp}{dt}

Try it yourself first

Hint: Think about how the force on the car relates to its charge and the electric field.

Step-by-step solution

  1. 1

    Identify forces

    Car has charge Q • Electric field E₀ • Force F = Q E₀
    F=QE0F = Q E_0
  2. 2

    Calculate total mass

    Initial mass M₀ • Mass added = n m t • Total mass M(t) = M₀ + n m t
    M(t)=M0+nmtM(t) = M_0 + n m t
  3. 3

    Apply Newton's second law

    Newton's 2nd law • F = dp/dt • p = M v
    F=ddt(M(t)v(t))F = \frac{d}{dt}(M(t)v(t))
  4. 4

    Solve differential equation

    Q E₀ = d/dt((M₀ + n m t)v) • Integrate both sides • Initial velocity = 0
    ∫QE0dt=∫d((M0+nmt)v)\int Q E_0 dt = \int d((M_0 + n m t)v)
  5. 5

    Final velocity

    Integrate and rearrange • v(t) = (Q E₀ t) / (M₀ + n m t)
    v(t)=QE0tM0+nmtv(t) = \frac{Q E_0 t}{M_0 + n m t}

Final answer

v(t) = (Q E₀ t) / (M₀ + n m t)

Remember: For variable mass systems, use F = dp/dt, not F = ma.

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Quick check · test yourself

A rocket expels exhaust gases at a constant rate. If the exhaust velocity relative to the rocket is constant, how does the thrust force change as the rocket's mass decreases?

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