Solving for t max in a 1D problem | Car speed, acceleration and timing involved

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AI Thread Summary
The discussion focuses on calculating the maximum time (t max) for a regular car traveling at a constant velocity (v0) to collide with a race car that accelerates from rest at a constant rate (a) after a stop light. The challenge lies in determining the relationship between the two cars' speeds and the timing of the race car's acceleration. The equations of motion provided are essential for solving the problem, but the lack of numerical values complicates the assignment of knowns and unknowns. It is noted that once the race car reaches the speed of the regular car, a collision becomes impossible. The key question remains how long it takes for the race car to reach the speed v0.
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Homework Statement


The general idea is a regular car moving at constant velocity (v0) is attempting to ram a race car whose wheels are spinning at a stop light. The race car waits until the last moment to push the gas pedal, accelerating at constant acceleration (a).
The time the race car starts to accelerate is t=0

I'm trying to find t max, basically after the race car accelerates, the longest time you have for the car (with Vo) to run into the race car.

Homework Equations


Vf=Vi + a(Tf-Ti)
Sf=Si + Vi(Tf-Ti) + 1/2a(Tf-Ti)^2
Vf^2=Vi^2 + 2a (Sf-Si)

The Attempt at a Solution


I'm having a bit of trouble assigning the knowns and unknowns due to no numbers being given, so I'm not sure how to start it.
 
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It seems like once the race car has reaches v0, it is impossible for the regular car to ram it. How long does it take for the race car to reach v0?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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