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oddjobmj
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Homework Statement
A cart, starting at rest, is below a hopper at t=0. The hopper drops coal into the cart at a constant rate μ. At t=0 a constant force starts pushing the cart to the right and the coal begins to fall. The force ends once the back of the cart passes the hopper.
A) Find final velocity of cart
B) Find total mass of coal delivered to the cart
Mass of empty cart = M = 3000 kg
Length of cart = L = 20 m
Mass delivery rate = μ = 507 kg/s
Force = F = 1343 N
Homework Equations
F=ma
The Attempt at a Solution
F=ma => a=[itex]\frac{F}{m}[/itex] = [itex]\frac{F}{m_0+μt}[/itex]=[itex]\frac{dv}{dt}[/itex]
Since there are no time dependent variables can I just do this?:
[itex]\int[/itex]dv=[itex]\frac{F dt}{m_0+μt}[/itex] => v=[itex]\frac{F*ln(m_0+μt)}{μ}[/itex]+C1
When t=0, v=0 so:
0=[itex]\frac{F*ln(m_0)}{μ}[/itex]+C1 => C1=[itex]\frac{-F*ln(m_0)}{μ}[/itex]
v(t)=[itex]\frac{F*ln(m_0+μt)}{μ}[/itex]-[itex]\frac{F*ln(m_0)}{μ}[/itex]=[itex]\frac{dx}{dt}[/itex]
[itex]\int[/itex]dx=[itex]\int[/itex][itex]\frac{F}{μ}[/itex](ln(m0+μt)-ln(m0))dt =>
x(t)=[itex]\frac{F(-μt-μt*ln(m_0)-(m_0+μt)*ln(m_0+μt)+m_0*ln(m_0))}{μ^2}[/itex]+C2
When t=0, x=0 so:
C2=[itex]\frac{Fm_0*ln(m_0)}{μ^2}[/itex]
Plugging that into the x(t) equation, solving for t, and plugging in the knowns I get
v(t)=2.9 m/s
I guess mass delivered is just μ*t=6000 kg
Does that look right? Where did I go wrong? Thanks for your help!