During a test run on an airport runway, a new race car reaches a speed of 258.4 mph from a standing start. The car accelerates with constant acceleration and reaches this speed mark at a distance of \(612.5 \mathrm{~m}\) from where it started. What was its speed after one-fourth, one-half, and three-fourths of this distance?

To convert the given speed from miles per hour (mph) to meters per second (m/s), we can use the following conversion factors: \(1 \mathrm{~mph} = \frac{1609.34}{3600} \mathrm{~m/s}\). Multiply the given speed (258.4 mph) by this conversion factor: \(258.4 \mathrm{~mph} \cdot \frac{1609.34}{3600} \mathrm{~m/s}\).

By multiplying the given speed by the conversion factor, we find the final speed in meters per second: \(v_f = 258.4 \mathrm{~mph} \cdot \frac{1609.34}{3600} \mathrm{~m/s} \approx 115.492 \mathrm{~m/s}\).

We can use the following kinematic equation to find the acceleration given the final speed \(v_f\), initial speed \(v_0\) (0 m/s in this case), distance \(d\), and acceleration \(a\): \(v_f^2 = v_0^2 + 2ad\). Since the car started from a standing position, \(v_0 = 0\). Solve for \(a\) to find the acceleration: \(a = \frac{v_f^2}{2d}\).

Plug the values for final speed and distance into the acceleration equation: \(a = \frac{(115.492)^2}{2(612.5)} \approx 10.86 \mathrm{~m/s^2}\).

Using the same kinematic equation from step 3, we can find the speeds at the three given distances by replacing \(d\) with the appropriate fractional distance (one-fourth, one-half, and three-fourths of 612.5 meters) and solving for the corresponding final speed value at each distance: 1. One-fourth of distance: \(v_{1/4}^2 = 2a(\frac{1}{4}d)\). Solve for \(v_{1/4}\): \(v_{1/4} = \sqrt{2(10.86)(153.125)} \approx 57.716 \mathrm{~m/s}\). 2. One-half of distance: \(v_{1/2}^2 = 2a(\frac{1}{2}d)\). Solve for \(v_{1/2}\): \(v_{1/2} = \sqrt{2(10.86)(306.25)} \approx 81.719 \mathrm{~m/s}\). 3. Three-fourths of distance: \(v_{3/4}^2 = 2a(\frac{3}{4}d)\). Solve for \(v_{3/4}\): \(v_{3/4} = \sqrt{2(10.86)(459.375)} \approx 99.862 \mathrm{~m/s}\). So, the car's speeds after one-fourth, one-half, and three-fourths of the distance are approximately \(57.716 \mathrm{~m/s}\), \(81.719 \mathrm{~m/s}\), and \(99.862 \mathrm{~m/s}\), respectively.