The conductor on a train spots a cow on the rails 200 meters ahead and immediately applies the brakes. The train decelerates at a constant deceleration \(a\) until it stops after 20 seconds, just before hitting the cow. Can the initial speed of the train in the instant before the brakes were applied be determined? (A) Yes, by dividing the given time by the given distance. (B) No, because to find the initial speed requires the deceleration, and the deceleration cannot be determined. (C) Yes, by solving for \(a\) in the equation \(2 a s=v_o^2-v_f^2\) and then using the equation \(s=-\left(\frac{1}{2}\right) a t^2+v_o t\) (D) Yes, by finding the displacement in terms of the average speed. (E) Both (C) and (D).

From the given problem, we know the following: - Final displacement (\(s\)) is 200 meters; - Time the train takes to stop (\(t\)) is 20 seconds; - The train has a constant deceleration \(a\); - We need to find the initial speed of the train (\(v_o\)).

Now we'll scrutinize each given option to see if it can help us determine the initial speed of the train. (A) Dividing the given time by the given distance is incorrect since it won't give us the initial speed \(v_0\). \ (B) At this point, we cannot conclude that \(a\) cannot be determined. So option (B) too early to choose. \ (C) The equation provided in this option, \(2as=v_o^2-v_f^2\), is indeed the formula representing a body's displacement having initial speed \(v_0\), final speed \(v_f\), and acceleration \(a\). In the given problem, it can be applied as follows: \ As the train comes to a stop, \(v_f = 0\). We can now solve for \(a\): \(2as = v_0^2 - v_f^2\) \(2as = v_0^2\) \(a = \frac{v_0^2}{2s}\) Now we can use the formula for displacement in terms of acceleration and initial velocity: \(s = -\frac{1}{2}at^2 + v_ot\) Now plug in \(\frac{v_0^2}{2s}\) for \(a\): \(s = -\frac{1}{2}\Big(\frac{v_0^2}{2s}\Big)t^2 + v_ot\) This expression can, in fact, be solved for \(v_o\). Thus, the answer (C) is a valid choice. \ (D) This option suggests using average speed to find the displacement. The average speed can be found using: Average speed = \(\frac{Initial \ velocity + Final \ velocity}{2}\) However, we do not have the initial speed and thereby it's impossible to find average speed. \ Therefore, option (D) isn't valid. Since there is no valid reason to mark (E) as it has option (D) included, we now choose (C) as the answer. So, the correct answer is (C).