Two objects, \(A\) and \(B\), collide. A has mass \(2.0 \mathrm{~kg}\), and \(B\) has mass \(3.0 \mathrm{~kg}\). The velocities before the collision are \(\overrightarrow{\mathbf{v}}_{\mathrm{iA}}=\) \((15 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(30 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{v}}_{\mathrm{i} B}=(-10 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\) After the collision, \(\overrightarrow{\mathbf{v}}_{\mathrm{fA}}=(-6.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(30 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\). What is the final velocity of \(B\) ?

Calculate the total initial momentum (before the collision) for the objects in both the \(\hat{i}\) and \(\hat{j}\) direction. This is done by adding up the momentum (mass × velocity) of the two objects in each direction. For object A in \(\hat{i}\) direction, it will be \(2.0 \mathrm{~kg} * 15 \mathrm{~m/s}\), and for object B, it will be \(3.0 \mathrm{~kg} * -10 \mathrm{~m/s}\). Similarly, calculate for \(\hat{j}\) direction.

Next, calculate the total final momentum of object A after the collision in both the \(\hat{i}\) and \(\hat{j}\) direction. This is again done by finding the momentum of object A in each direction using the given final velocity. For the \(\hat{i}\) direction it will be \(2.0 \mathrm{~kg} * -6.0 \mathrm{~m/s}\) and for the \(\hat{j}\) direction it is \(2.0 \mathrm{~kg} * 30 \mathrm{~m/s}\).

Finally, calculate the final velocity \(\overrightarrow{\mathbf{v}}_{\mathrm{fB}}\) in both directions. According to the law of conservation of momentum, the total initial momentum (before collision) should equal the total final momentum (after collision). So, set up equations equating total initial and final momentum in \(\hat{i}\) direction and \(\hat{j}\) direction separately. For object B's final velocity in the \(\hat{i}\) direction, \(\overrightarrow{\mathbf{v}}_{\mathrm{fBi}}\), subtract the final momentum of object A in the \(\hat{i}\) direction from the total initial \(\hat{i}\) momentum, and divide it by the mass of object B. Do similar calculation for the \(\hat{j}\) direction, \(\overrightarrow{\mathbf{v}}_{\mathrm{fBj}}\), to get the resultant final velocity (in vector form) of object B after the collision.