To analyze the motion of the rocket sled from the perspective of an observer, we can break down the problem into two parts: first, we need to establish the relationship between the angular position of the rocket sled and time, and then we will determine when the sled's motion becomes unfeasible. Let’s tackle each part step-by-step.

Part (a): Finding the Angular Position Function

Given that the rocket sled moves along a straight horizontal track with a constant velocity, let’s denote the following variables:

• v: the constant velocity of the rocket sled
• b: the distance from the observer to the track
• x(t): the position of the rocket sled at time t, which can be expressed as x(t) = vt

The angular position θ of the sled relative to the observer can be found using the tangent function, where:

tan(θ) = x(t) / b

Substituting the expression for x(t), we have:

tan(θ) = (vt) / b

To find θ, we take the arctangent:

θ(t) = arctan(vt/b)

This equation describes how the angular position of the sled changes over time, starting at θ(0) = 0 when t = 0, which is the moment the sled is closest to the observer.

Part (b): Determining the Time When Motion Becomes Impossible

Next, we need to find the time tc at which the motion of the sled becomes physically impossible. This situation occurs when the sled reaches a point where the observer can no longer track its motion effectively. One possible interpretation could be when the sled travels far enough that the angle approaches a vertical position (90 degrees). In terms of the tangential function, this happens when:

tan(θ) → ∞

Thus, we need to set the argument of the tangent function to be undefined:

vt/b → ∞

This condition implies that as t approaches infinity, the sled's position x(t) will keep moving away from the observer without bounds. However, we need to consider practical limits. For instance, if we consider the limits of tracking equipment, we might say that tc is the time when the sled is at its maximum observable angle. If we set a maximum observability angle, say θ_max, we can find tc by rearranging our earlier equation:

tan(θ_max) = (v * tc) / b

Rearranging gives:

tc = (b * tan(θ_max)) / v

In summary, the specific time tc will depend on the chosen maximum angle θ_max and the sled's velocity. If we assume θ_max is a significant angle, such as 89 degrees, we can calculate tc using the aforementioned formula. Remember, as angles approach 90 degrees, tracking becomes increasingly difficult, leading to practical limitations in observing the sled's motion.

Wrapping Up

To summarize, we derived the angular position function θ(t) = arctan(vt/b) for the rocket sled moving with constant velocity and identified the conditions under which tracking becomes impossible, expressed through the equation tc = (b * tan(θ_max)) / v. This analysis highlights the interplay between the sled's motion and the observer's ability to track it effectively.