91Ó°ÊÓ

Velocity is a measure of how fast an object is moving and in what direction. It's calculated using the formula:

• \( v = u + at \)

where:

• \( v \) is the final velocity,
• \( u \) is the initial velocity,
• \( a \) is the acceleration,
• \( t \) is the time over which the acceleration occurs.

In our speedboat problem, the boat starts from rest, meaning \( u = 0 \) for the first interval. The boat's velocity increases due to acceleration. After multiple intervals of acceleration, both positive and negative, the final velocity can be calculated after each stage by adding the previous velocity to the product of acceleration and time for that interval.
For instance, after the first phase, using \( u = 0 \), \( a = 2.01 \, \mathrm{m/s^2} \), and \( t = 7.00 \, \mathrm{s} \), the velocity becomes \( 14.07 \, \mathrm{m/s} \). This process repeats for subsequent stages of acceleration.
By understanding and applying this formula, we can calculate how the velocity changes over different phases of motion.

Acceleration is how quickly an object changes its velocity. It can be a result of speeding up, slowing down, or changing direction. It's crucial to differentiate between positive acceleration (speeding up) and negative acceleration (slowing down), often called deceleration.
The formula to determine the final velocity given an initial velocity and acceleration is:

• \( v = u + at \)

In our scenario:

• First, the boat accelerates positively at \( 2.01 \, \mathrm{m/s^2} \).
• Then, it continues with a reduced positive acceleration of \( 0.518 \, \mathrm{m/s^2} \).
• Finally, the boat experiences a negative acceleration or deceleration of \( -1.49 \, \mathrm{m/s^2} \).

The overall behavior of the speedboat is shaped by these different accelerations. Acceleration affects not just the speed but also the displacement and how far the boat travels during each phase. Keeping track of the sign of the acceleration is crucial, as it informs whether the velocity increases or decreases with time.

Displacement refers to the change in position of an object. It's not just about how far an object traveled but also its direction. The formula used to find the displacement during an interval of accelerated motion is:

• \( s = ut + \frac{1}{2}at^2 \)

where:

• \( s \) is the displacement,
• \( u \) is the initial velocity,
• \( a \) is the acceleration,
• \( t \) is the time duration of motion.

For our speedboat, this equation is applied three times, each for different phases with varying accelerations. By summing these calculated displacements:

• First part: \( 49.245 \, \mathrm{m} \)
• Second part: \( 90.726 \, \mathrm{m} \)
• Third part: \( 71.424 \, \mathrm{m} \)

we find the total displacement of the boat. Thus, the total displacement, which represents how far from its start the boat finally is, calculates as \( 211.395 \, \mathrm{m} \). Displacement is essential for understanding the actual journey and final position of an object in motion.