Solved by AI teacher · Physics

Find velocity of car/tank as fx of time.

Concept: Variable mass system

Question

F = \frac{dp}{dt} = m \frac{dv}{dt} + v \frac{dm}{dt}

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Hint: Think about how the total mass of the car/tank changes over time.

1
Identify forces
Force F = Q E₀
$F = Q E_0$
2
Mass at time t
Initial mass M₀ • Mass added = n m t • Total mass M(t) = M₀ + n m t
$M(t) = M_0 + n m t$
3
Apply Newton's 2nd Law
F = dp/dt • F = M dv/dt + v dM/dt • dM/dt = n m
$F = M \frac{dv}{dt}$
4
Solve differential equation
Q E₀ = (M₀ + n m t) dv/dt • dv = (Q E₀ / (M₀ + n m t)) dt • Integrate both sides
$\int dv = \int \frac{Q E_0}{M_0 + n m t} dt$
5
Final velocity
v(t) = (Q E₀ / (n m)) ln(M₀ + n m t) - (Q E₀ / (n m)) ln(M₀) • v(t) = (Q E₀ / (n m)) ln((M₀ + n m t) / M₀)
$v(t) = \frac{Q E_0}{n m} \ln\left(\frac{M_0 + n m t}{M_0}\right)$

v(t) = (Q E₀ t) / (M₀ + n m t)

Remember: Variable mass systems need careful application of Newton's second law.

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In a variable mass system, if the external force is zero and mass is being ejected, what happens to the velocity of the main body?

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