91Ó°ÊÓ

Kinematics is the branch of physics that describes the motion of objects without considering the causes of this motion, such as forces. It deals with various parameters like displacement, velocity, and acceleration.
In this collision problem, the passenger train and the freight train each have distinct kinematics. The passenger train has a certain initial velocity, an acceleration applied against its direction, and it moves according to its position-time equation
\[ x_p(t) = x_{p0} + v_{p0}t + \frac{1}{2}a_pt^2 \].
The freight train, on the other hand, moves with a constant speed, so its position-time relation is linear:
\[ x_f(t) = x_{f0} + v_{f}t \].
Understanding these equations is crucial, as they allow us to plot each train’s journey over time, helping us predict where and when they might collide.
These two equations are mathematical models of their respective kinematics, representing how their positions change over time.

When analyzing the collision of two trains moving with different equations of motion, we end up needing to solve a quadratic equation. This kind of equation appears in scenarios where two objects keep changing their relative distance over time, such as with accelerating and constant-speed motion.
The quadratic equation resulting from this problem has the form:
\[ -0.05t^2 + 10t - 200 = 0 \]
which can be solved using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
Here, \( a = -0.05 \), \( b = 10 \), and \( c = -200 \).
Upon solving, we find two potential collision times. It's important to choose the realistic one, where time can only be positive, yielding that the feasible collision time is \( t = 36.8 \ \text{s} \).
The discriminant \( b^2 - 4ac \) in this context tells us the nature of the roots. A positive discriminant indicates two real roots, aligning with the physical scenario of collision prediction.

A position-time graph is a powerful tool in physics for visualizing the motion of objects. In this collision scenario, such a graph illustrates how the position of each train changes over time.
To sketch this graph, the horizontal axis represents time \( t \) and the vertical axis stands for position \( x \).
The passenger train's path is a curve due to its acceleration opposite to its motion. Its graph follows the equation:
\[ x_p(t) = 25.0t - 0.05t^2 \],
showcasing a parabola that decreases over time as the train slows down.
The freight train moves at a constant speed, its graph is a straight line:
\[ x_f(t) = 200 + 15t \],
with a slope representing its constant velocity.
The collision point is where these two paths intersect. On the graph, it occurs at the position \( x = 752 \ \text{m} \) when \( t = 36.8 \ \text{s} \).
Drawing such graphs allows us to visually analyze and understand the motion dynamics involved in kinematics problems like collisions.