- #1
Nate Stevens
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Homework Statement
You throw baseballs at a mass rate of σ kg/s (assume a continuous rate) with speed u at the back of a car. They collide elastically with the car. The car starts at rest and has mass M. It moves on the ground with no friction. Find its speed and position as a function of time.
Homework Equations
F=ma=mdv/dt=dp/dt
The Attempt at a Solution
To start, I thought I should consider a single collision between one baseball and the car after it has started moving.
I let the thrown baseball have a mass of dm. Then when it hits the car, the car will only gain an infinitesimal amount of speed.
So the initial velocities before the collision are
baseball -> u
car -> v
And the final velocities after the collision are
baseball -> ??
car -> v + dv
If I use the fact that in elastic collisions the relative velocities of two objects (before and after the collision) are reversed, then
u-v = - ( ??-( v+dv ))
solving for ??
?? = 2v+dv-u
Now that we have the baseball's initial and final velocity we can get it's change in momentum
Δp = pf - pi = (2v+dv-u)×dm - (u)×dm = dm×dv + (2v-2u)×dm
I know from here I can get the force on the ball by differentiating the above momentum with respect to t, but before that I need to figure out what to do with the dm×dv. Is it okay say that this will go to zero since it is two very small values multiplied together?
If that is the case, are the final steps just rewriting dm/dt as σ, plugging the momentum into
mdv/dt=dp/dt, and separating variables twice? Somehow I don't think so, since σ is just the mass rate of baseballs being thrown and NOT the mass rate of baseballs hitting the car.
Even if it is just answering the dm*dv question, any help would really be appreciated.