A sports car traveling along a straight line increases its speed from $20.0 \mathrm{mi} / \mathrm{h}\( to \)60.0 \mathrm{mi} / \mathrm{h} .$ (a) What is the ratio of the final to the initial magnitude of its momentum? (b) What is the ratio of the final to the initial kinetic energy?

To solve this problem consistently, we need to convert velocities from miles per hour \(\mathrm{mi} / \mathrm{h}\) to meters per second \(\mathrm{m} / \mathrm{s}\). Using the conversion factor, 1 mile = 1609.34 meters and 1 hour = 3600 seconds, we obtain the initial and final velocities in SI units. Initial velocity (\(v_1\)): \(20.0\frac{\mathrm{mi}}{\mathrm{h}}\times\frac{1609.34\mathrm{m}}{1\mathrm{mi}}\times\frac{1\mathrm{h}}{3600\mathrm{s}}\approx8.94\frac{\mathrm{m}}{\mathrm{s}}\) Final velocity (\(v_2\)): \(60.0\frac{\mathrm{mi}}{\mathrm{h}}\times\frac{1609.34\mathrm{m}}{1\mathrm{mi}}\times\frac{1\mathrm{h}}{3600\mathrm{s}}\approx26.82\frac{\mathrm{m}}{\mathrm{s}}\)

First, we find the initial and final momentum magnitudes using the formula \(p = m*v\). Then, we calculate the ratio of final to initial momentum. We will not use actual mass, as it will cancel out in the ratio. Initial momentum magnitude (\(p_1\)): \(m * 8.94\frac{\mathrm{m}}{\mathrm{s}}\) Final momentum magnitude (\(p_2\)): \(m * 26.82\frac{\mathrm{m}}{\mathrm{s}}\) Ratio of final to initial momentum: \(\frac{p_2}{p_1}=\frac{m * 26.82}{m * 8.94}=\frac{26.82}{8.94}\approx3\)

First, we find the initial and final kinetic energy magnitudes using the formula \(KE = \frac{1}{2}mv^2\). Then, we calculate the ratio of final to initial kinetic energy. Again, we will not use actual mass, as it will cancel out in the ratio. Initial kinetic energy magnitude (\(KE_1\)): \(\frac{1}{2}m * (8.94\frac{\mathrm{m}}{\mathrm{s}})^2\) Final kinetic energy magnitude (\(KE_2\)): \(\frac{1}{2}m * (26.82\frac{\mathrm{m}}{\mathrm{s}})^2\) Ratio of final to initial kinetic energy: \(\frac{KE_2}{KE_1}=\frac{\frac{1}{2}m * (26.82\frac{\mathrm{m}}{\mathrm{s}})^2}{\frac{1}{2}m * (8.94\frac{\mathrm{m}}{\mathrm{s}})^2}=\frac{(26.82\frac{\mathrm{m}}{\mathrm{s}})^2}{(8.94\frac{\mathrm{m}}{\mathrm{s}})^2}\approx9\) #Conclusion#: (a) The ratio of the final to the initial magnitude of its momentum : 3 (b) The ratio of the final to the initial kinetic energy: 9