In waterskiing, a "garage sale" occurs when a skier loses control and falls and waterskis fly in different directions. In one particular incident, a novice skier was skimming across the surface of the water at \(22.0 \mathrm{~m} / \mathrm{s}\) when he lost control. One ski, with a mass of \(1.50 \mathrm{~kg},\) flew off at an angle of \(12.0^{\circ}\) to the left of the initial direction of the skier with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). The other identical ski flew from the crash at an angle of \(5.00^{\circ}\) to the right with a speed of \(21.0 \mathrm{~m} / \mathrm{s} .\) What was the velocity of the \(61.0-\mathrm{kg}\) skier? Give a speed and a direction relative to the initial velocity vector.

Before falling, the skier is moving at a velocity of \(22.0 m/s\). The mass of the skier is \(61.0 kg\). Therefore, we can calculate the initial momentum of the skier as follows: Initial momentum of skier = mass * velocity = \(61.0 kg \times 22.0 m/s = 1342 kg\cdot m/s\)

Let's compute the momenta of the flying water skis: Both skis have a mass of \(1.50 kg\). The first ski is flying at \(25.0 m/s\) at an angle of \(12.0^\circ\) to the left and the second ski is flying at \(21.0 m/s\) at an angle of \(5.0^\circ\) to the right of the initial direction of the skier. To find the momentum of each ski, we have to resolve the velocity vector of each ski into its horizontal and vertical components. - For the first ski: Horizontal component = \(25 \cos(12^\circ)\) Vertical component = \(25 \sin(12^\circ)\) - For the second ski: Horizontal component = \(21 \cos(5^\circ)\) Vertical component = \(21 \sin(5^\circ)\) The momentum of each ski can then be computed as follows: - Momentum of the first ski = \((1.5 kg)(25\cos(12^\circ)\hat{i} + 25\sin(12^\circ)\hat{j})\) - Momentum of the second ski = \((1.5 kg)(21\cos(5^\circ)\hat{i} + 21\sin(5^\circ)\hat{j})\)

The total initial momentum equals the total final momentum. That is, the initial momentum of the skier is equal to the sum of the final momentum of the skier and the momenta of the flying water skis. Therefore, the final momentum of the skier can be calculated as follows: Final momentum of skier = Initial momentum of skier - (momentum of first ski + momentum of second ski) Final momentum of skier = \(1342\hat{i} kg\cdot m/s - [(1.5 kg)(25\cos(12^\circ)\hat{i} + 25\sin(12^\circ)\hat{j}) + (1.5 kg)(21\cos(5^\circ)\hat{i} + 21\sin(5^\circ)\hat{j})]\) Now, calculate the horizontal and vertical components of the final momentum of the skier.

Now that we have the final momentum of the skier, we can find the final velocity of the skier by dividing the final momentum by the mass of the skier: Final velocity of skier = \(\frac{Final~momentum~of~skier}{mass~of~skier}\) = \(\frac{Final~momentum~of~skier}{61.0 kg}\) To find the final speed of the skier, we will find the magnitude of the final velocity vector: Speed of the skier = \(|\textrm{Final velocity of skier}| = \sqrt{(Final~Horizontal~Component)^2 + (Final~Vertical~Component)^2}\) To find the direction of the skier with respect to the initial velocity vector, we use the inverse tangent (arctan) function: Direction of the skier = \(tan^{-1}\left(\frac{Final~Vertical~Component}{Final~Horizontal~Component}\right)\) Calculate the final speed and direction of the skier using the computed values.