91Ó°ÊÓ

In physics, the concept of uniformly accelerated motion describes an object that is speeding up or slowing down at a steady rate. When the acceleration is constant, the motion of the object can be analyzed using a set of equations known as the kinematic equations.

One of these is the equation for the displacement during a period of constant acceleration, which is given by \(d = ut + \frac{1}{2}at^2\), where \(d\) is the displacement, \(u\) is the initial velocity, \(t\) is the time, and \(a\) is the acceleration. Since both the man and the horse in the given problem are starting from rest (u = 0 m/s), the formula simplifies to \(d = \frac{1}{2}at^2\). This equation allows us to calculate the distance each competitor covers during their acceleration phase before reaching a constant speed.

The uniform acceleration gives us a predictable pattern of motion, making it easier to calculate various aspects of motion, which is valuable for solving kinematics problems.

Calculating the final speed of an object after a period of uniform acceleration is crucial for understanding the dynamics of its motion. The formula to find the final velocity \(v\) after acceleration is \(v = u + at\), where \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time during which the object accelerated.

When the motion starts from rest, as with the man and the horse in our race scenario, the initial velocity \(u\) is zero and the equation simplifies to \(v = at\). By inputting the respective acceleration and time values for the man and the horse into this equation, we determine their final speeds after the period of acceleration. This is the speed they will maintain after they stop accelerating and continue with a constant velocity towards the finish line.

The relationship between time and distance is a fundamental aspect of kinematics. Once we know the speed at which an object is moving (assuming it's constant), we can calculate the time it takes to cover a certain distance with the simple equation \(t = \frac{d}{v}\), where \(t\) is the time, \(d\) is the distance, and \(v\) is the velocity. When either the distance or the time is unknown, rearranging this equation can solve for the missing variable.

Looking at our race, after the man and the horse have each reached their constant speeds, we can apply this equation to find out how long it takes them to complete the remaining distance to the finish line. In practice, we use the speeds calculated after acceleration and the remaining distances, which we obtained by subtracting the distances they covered while accelerating from the total of 200 meters for the horse and 100 meters for the man, allowing us to calculate the time each will take to reach the finish, and thus, determine who wins the race.