Runner's Deceleration: Calculate Acceleration & Velocity

by Dimemap Team 57 views

Let's break down a classic physics problem involving a runner who unfortunately takes a tumble! We're going to calculate a few key things: their average acceleration, average velocity, and the total distance they covered while slowing down. So, lace up your thinking shoes, and let's dive in!

Understanding the Scenario

Imagine a runner cruising along at a steady 5 m/s. Suddenly, disaster strikes – they trip! From that moment on, they steadily lose speed, decelerating to 2 m/s over a span of 10 seconds before hitting the ground. Our mission is to figure out exactly how their motion changed during those fateful 10 seconds. We'll be using some fundamental physics concepts to get there, guys.

a) Calculating Average Acceleration

Average acceleration is all about how quickly the velocity changes over a certain period. It's not about the instantaneous acceleration at a specific moment but the overall change throughout the motion. The formula for average acceleration is super straightforward:

  • a = (vf - vi) / t

Where:

  • a is the average acceleration
  • vf is the final velocity
  • vi is the initial velocity
  • t is the time interval

In our runner's case:

  • vi = 5 m/s (the initial speed when they tripped)
  • vf = 2 m/s (their speed right before falling)
  • t = 10 s (the time it took to decelerate)

Plugging these values into the formula, we get:

  • a = (2 m/s - 5 m/s) / 10 s = -3 m/s / 10 s = -0.3 m/s²

So, the average acceleration is -0.3 m/s². The negative sign tells us that the runner was decelerating, meaning their velocity was decreasing. This makes perfect sense since they were slowing down before the fall. Acceleration, in this context, is the rate at which the runner's speed diminished, and the negative value simply indicates the direction of this change – opposite to the direction of motion. This calculation provides a clear and concise measure of how quickly the runner lost speed during those critical 10 seconds, giving us a quantitative understanding of their deceleration.

b) Determining Average Velocity

Now, let's find the average velocity during those 10 seconds. Average velocity isn't quite the same as simply averaging the initial and final speeds, especially if the acceleration isn't constant (though in this case, it is constant, simplifying things). However, for constant acceleration, we can use a simple formula:

  • v_avg = (vi + vf) / 2

Where:

  • v_avg is the average velocity
  • vi is the initial velocity
  • vf is the final velocity

Using the values from our problem:

  • vi = 5 m/s
  • vf = 2 m/s

Therefore:

  • v_avg = (5 m/s + 2 m/s) / 2 = 7 m/s / 2 = 3.5 m/s

Thus, the average velocity of the runner during those 10 seconds is 3.5 m/s. This means that, on average, the runner was moving at this speed during the deceleration period. It's a simple average of the starting and ending speeds because the deceleration was constant. Knowing the average velocity helps us understand the overall pace at which the runner was moving during this time, providing a valuable piece of the puzzle as we analyze their motion. Furthermore, it serves as a crucial component in calculating the total distance covered, which we'll tackle in the next section.

c) Calculating the Distance Traveled

To figure out the distance the runner covered while slowing down, we can use one of the kinematic equations. Since we know the initial velocity, final velocity, acceleration, and time, we have a few options. A convenient one is:

  • d = vi * t + 0.5 * a * t²

Where:

  • d is the distance traveled
  • vi is the initial velocity
  • t is the time
  • a is the acceleration

Let's plug in our values:

  • vi = 5 m/s
  • t = 10 s
  • a = -0.3 m/s²

So:

  • d = (5 m/s * 10 s) + (0.5 * -0.3 m/s² * (10 s)²) = 50 m + (0.5 * -0.3 m/s² * 100 s²) = 50 m - 15 m = 35 m

Therefore, the runner traveled 35 meters while decelerating before falling. This tells us the total ground covered during those 10 seconds of slowing down. It's a practical measure that combines the effects of initial speed, deceleration rate, and time. Understanding the distance traveled provides a complete picture of the runner's motion, allowing us to visualize the scenario and comprehend the physical consequences of their fall. This final calculation wraps up our analysis, giving us a comprehensive understanding of the runner's deceleration.

Summary of Results

Alright, guys, let's recap what we've found:

  • Average Acceleration: -0.3 m/s² (deceleration)
  • Average Velocity: 3.5 m/s
  • Distance Traveled: 35 m

These calculations paint a clear picture of the runner's motion during those critical 10 seconds. We've successfully determined how quickly they slowed down, their average speed, and how far they traveled before hitting the ground. Physics for the win! Understanding these concepts helps us analyze and predict motion in various real-world scenarios. Keep practicing, and you'll be a physics pro in no time!