Let's let the standard cart be S, little i be "initial" and little f be "final." Conserve momentum for the initial collision:
mS*vSi + mA*vAi = mS*vSf + mA*vAf
1.0kg*0.40m/s + mAi*0m/s = 1.0kg*0.20m/s + mA*0.70m/s
mA = 0.2857 kg
Now conserve momentum for the collision of the putty (P) with cart A. Little c means "center" (collision), and the final velocities from the previous collision are the initial velocities here:
mA*vAi + mP*vPi = (mA + mP)*vAc
0.2857kg*-0.40m/s + mP*0m/s = (0.2857kg + mP)*vAc
0.2857kg + mP = -0.1143kg·m/s / vAc ← equation #1
And finally, conserve momentum for the final collision; P indicates the putty:
mS*vSc + (mA+mP)*vAc = mS*vSf + (mA+mP)*vAf
1.0kg*0.20m/s + (0.2857kg + mP)*vAc = 1.0kg*-0.20m/s + (0.2857kg + mP)*0.45m/s
substitute for (0.2857kg + mP) (equation #1) and do some multiplication:
0.20kg·m/s - (0.1143kg·m/s / vAc) * vAc = -0.20kg·m/s - (0.1143kg·m/s / vAc)*0.45m/s
0.0857 kg·m/s = -0.20kg·m/s - 0.051435kg·m²/s² / vAc)
0.2857 kg·m/s = -0.051435kg·m²/s² / vAc
vAc = -0.051435kg·m²/s² / 0.2857kg·m/s = -0.18 m/s
plug into equation #1
0.2857kg + mP = -0.1143kg·m/s / -0.18m/s = 0.6349 kg
mP = 0.35 kg ◄
Not extremely difficult, just a nuisance keeping track of the three collisions.
Also note that one might have set up equation #1 to be
vAc = -0.1143kg·m/s / (0.2857kg + mP)
but I thought that would lead to slightly nastier math. It would, however, lead so a solution for mP without having to solve for vAc.
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