Question

Starting from rest, a boat increases its speed to \(5.00 \mathrm{~m} / \mathrm{s}\) with constant acceleration. a) What is the boat's average speed? b) If it takes the boat 4.00 s to reach this speed, how far has it traveled?

Short answer

Question: A boat accelerates from rest to a speed of 5.00 m/s with constant acceleration. If it takes 4.00 seconds to reach this speed, find a) The boat's average speed, and b) The distance the boat travels. Answer: a) The boat's average speed is 2.50 m/s. b) The boat travels a distance of 10.00 meters.

Step-by-step solution

Calculate the average speed

The initial speed of the boat is 0 m/s (as it starts from rest), and the final speed is given as 5.00 m/s. We can calculate the average speed using the formula: Average speed = (Initial speed + Final speed) / 2 Average speed = (0 + 5.00) / 2 Average speed = 2.50 m/s The boat's average speed is 2.50 m/s.

Calculate the acceleration

We can calculate the acceleration using the formula: Acceleration = (Final speed - Initial speed) / time We are given that it takes the boat 4.00 seconds to reach the final speed of 5.00 m/s, so: Acceleration = (5.00 - 0) / 4.00 Acceleration = 5.00 / 4.00 Acceleration = 1.25 m/s² The boat's constant acceleration is 1.25 m/s².

Calculate the distance traveled

Now, we can find the distance traveled, using the formula: Distance = Initial speed × time + 0.5 × acceleration × time² Here, Initial speed = 0 m/s, acceleration = 1.25 m/s², and time = 4.00 s. Substitute the values in the formula: Distance = 0 × 4.00 + 0.5 × 1.25 × (4.00)² Distance = 0 + 0.5 × 1.25 × 16.00 Distance = 0.5 × 1.25 × 16.00 Distance = 10.00 m The boat travels a distance of 10.00 meters.

Key concepts

Average Speed Calculation

Understanding how to calculate average speed is essential in kinematics to describe how fast an object moves over a given period. The average speed gives us a simple way to quantify movement by considering the total distance traveled and the total time taken, without the complexities of accounting for variations in speed during the trip.

For an object starting from rest, like the boat in our example, the initial speed is zero. To get the average speed, we simply add the initial speed to the final speed and divide by two. Mathematically represented as:
\[\begin{equation}\text{Average speed} = \frac{\text{Initial speed} + \text{Final speed}}{2}\end{equation}\]
When the final speed is given, and the initial speed is zero, the calculation becomes even simpler, illustrating a foundational concept in kinematics: the mean value of two numbers.

Constant Acceleration

The concept of constant acceleration is a cornerstone in kinematics, especially when dealing with problems involving objects starting from rest or in uniform motion. Acceleration is defined as the rate at which the velocity of an object changes with time.

In cases where acceleration is constant, the formula we use is:
\[\begin{equation}\text{Acceleration} = \frac{\text{Final speed} - \text{Initial speed}}{\text{Time}}\end{equation}\]
For objects starting from a standstill, with an initial speed of zero, the formula simplifies considerably. With constant acceleration, the motion of an object can be easily predicted, emphasizing a predictable linear relationship between velocity and time, which forms a basis for more complex motion equations.

Distance Traveled

Calculating the distance traveled by an object under the influence of constant acceleration involves another core principle in kinematics. This value is essential for understanding the space an object covers as it moves. The general equation for this distance is sized to account for the initial velocity, time of travel, and acceleration:
\[\begin{equation}\text{Distance} = \text{Initial speed} \times \text{time} + 0.5 \times \text{acceleration} \times \text{time}^2\end{equation}\]
When starting from rest, the initial speed is zero, meaning the first term of the equation drops away, leaving us with the product of half the acceleration and the square of the time. This simplification highlights the quadratic relationship between distance and time under constant acceleration.