Two trucks of mass $M$ each are moving in opposite direction on adjacent parallel tracks with same velocity $u$ . One is carrying potatoes and other is carrying onions, a bag of potatoes has a mass $m_{1}$ and bag of onions has mass $m_{2}$ (included in the mass of truck $M$ ). When trucks get close to each other while passing, the drivers exchange a bag with the other one by throwing the other one. Find the final velocities of the trucks after exchange of the bags.

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Solution

Here at the time of exchange of bags, momentum in the direction of individual motion remains conserved. The situation is shown in the figure.
When driver of the first truck carrying potatoes, throws a bag of mass $m_{1}$ in a direction perpendicular to the motion direction, towards the other truck, as shown in the figure. During throw, the bag has a velocity $u$ in the direction of motion of the first truck. Similarly, when the second truck driver throws the onion bag conserve momentum for both the trucks independently as
First truck carrying potatoes
$m u - m_{1} u - m_{2} u = (M - m_{1} + m_{2}) ν_{1}$
On solving, velocity of the first truck after exchange is
$ν_{1} = \frac{M u - m_{1} u - m_{2} u}{M - m_{1} + m_{2}}$
Second truck carrying onions
$M u - m_{2} u - m_{1} u = (M - m_{2} + m_{1}) ν_{2}$
On solving, velocity of the first truck exchange is
$ν_{2} = \frac{M u - m_{1} u - m_{2} u}{M - m_{2} + m_{1}}$

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Two trucks of mass M each are moving in opposite direction on adjacent parallel tracks with same velocity u. One is carrying potatoes and other is carrying onions, bag of potatoes has a mass $m_{1}$ and bag of onions has a mass $m_{2}$ (included in the mass of truck M). When trucks get close to each other while passing, the drivers exchange a bag with the other one by throwing the other one. Find the final velocities of the trucks after exchange of the bags.