91Ó°ÊÓ

When dealing with uniform acceleration, understanding average speed is crucial. Average speed isn't just about finding the midpoint between speeds. It's actually the total distance traveled divided by the total time of travel.
This gives you a helpful way to think about how fast something is moving over a complete journey rather than at just a single instance.

• In the case of uniform (or constant) acceleration, calculating average speed gets a bit simpler because you're dealing with a straight line: the object starts moving from rest to its final speed in a straight acceleration pattern.

For our problem, since the boat starts from rest and accelerates uniformly to a speed of 4.12 m/s, the average speed can be easily calculated.
You have:
\[ \text{Average Speed} = \frac{v_0 + v}{2} \]
Substituting the given values, with initial speed \(v_0 = 0 \) m/s and final speed \(v = 4.12 \) m/s, we calculate:
\[ \text{Average Speed} = \frac{0 + 4.12}{2} = 2.06 \text{ m/s} \]
This tells us that over the period of acceleration, the boat's average speed is 2.06 meters per second.

Calculating how far something has traveled when it's accelerating isn't as tricky as it might seem at first. Since we've already found the average speed, the task becomes straightforward.
We simply need to use the formula relating distance, average speed, and time:

• \[ \text{Distance} = \text{Average Speed} \times \text{Time} \]

This helps us find how much ground an object has covered over a period of constant acceleration.
In our boat example, where the average speed is 2.06 m/s and the time is 4.77 seconds, we can find the distance:
\[ \text{Distance} = 2.06 \times 4.77 \]
After doing the math, we find:
\[ \text{Distance} = 9.8262 \text{ meters} \]
When you round this to a reasonable number of significant figures, it gives you approximately 9.83 meters.
This tells us the total distance the boat has traveled while accelerating to its final speed.

Understanding final velocity when dealing with uniform acceleration means recognizing the endpoint of an object's motion during a given time frame.
Final velocity is the speed of an object at the end of its acceleration period.

• In uniform acceleration, you start from an initial state and accelerate to reach this final speed.

In this case, the process is given extra clarity because we know the initial velocity and the boat accelerates to this final speed steadily over time.
The significant thing about final velocity in this problem is that it represents how quickly the boat is moving forward after accelerating from rest.
Knowing the final velocity \( v = 4.12 \text{ m/s} \) helps to set the context for understanding both average speed and the distance calculation.
It's the end result of this movement and frames our entire understanding of the boat's journey in this exercise.