The Thrilling Journey of a Speedboat: Reaching the Buoy in Style

When a speedboat with an initial speed of 31 m/s approaches a no-wake buoy marker located 100 m ahead, the pilot slows down the boat with a constant acceleration of -4.00 m/s². To calculate the time it takes for the speedboat to reach the buoy, we first make use of the formula ut + (1/2)at² = s, where u is the initial velocity, a is the acceleration (in this case, deceleration), t is the time taken, and s is the distance to the buoy.

Calculating the Time:

Given: Initial speed, u = 31 m/s Acceleration, a = -4 m/s² Distance to the buoy, s = 100 m

Substitute the values into the formula: (31 m/s) * t + (1/2) * (-4 m/s²) * t² = 100 m

This equation can be rearranged to: 2t² - 31t + 100 = 0

By solving this quadratic equation, we find two possible solutions for t: t = 10.92 seconds or t = 4.58 seconds

Since the velocity needs to be positive, the time taken for the speedboat to reach the buoy is 4.58 seconds.

Calculating the Velocity at the Buoy:

With the time known, we can determine the velocity of the speedboat when it reaches the buoy. The formula v = u + at can be used for this calculation.

Using the time t = 4.58 seconds, we find: v = 31 m/s - 4 m/s² * 4.58 s v = 12.68 m/s

Therefore, the final answer is: (a) The time it takes for the speedboat to reach the buoy is 4.6 seconds. (b) The velocity of the speedboat when it reaches the buoy is 12.7 m/s.

So, the speedboat successfully reaches the buoy in 4.6 seconds and maintains a velocity of 12.7 m/s, showcasing a thrilling journey filled with swift calculations and precise timing.