Problem 13 - An Introduction to Physical Science

Question

A motorboat starting from rest travels in a straight line on a lake. (a) If the boat achieves a speed of \(8.0 \mathrm{m} / \mathrm{s}\) in \(10 \mathrm{s}\), what is the boat's average acceleration? (b) If the boat continues with the same acceleration, what would be its speed after another \(5.0 \mathrm{s}\) ?

Step-by-Step Solution

Answer

(a) The average acceleration is \(0.8 \, \mathrm{m/s^2}\). (b) The speed after another \(5.0\, \mathrm{s}\) is \(12.0\, \mathrm{m/s}\).

Step 1: Understanding the Problem

Determine the initial speed (\(u\)), final speed (\(v\)), time (\(t\)), and acceleration (\(a\)) for the situation described. Initially, \(u = 0\, \mathrm{m/s}\) since the boat starts from rest. The problem provides the final speed \(v = 8.0 \, \mathrm{m/s}\) after \(t = 10 \, \mathrm{s}\).

Step 2: Calculate Average Acceleration

Use the formula for acceleration, \(a = \frac{v - u}{t}\). Substitute the known values: initial speed \(u = 0\, \mathrm{m/s}\), final speed \(v = 8.0 \, \mathrm{m/s}\), and time \(t = 10 \, \mathrm{s}\). Thus, \(a = \frac{8.0 - 0}{10} = 0.8 \, \mathrm{m/s^2}\).

Step 3: Extend the Motion for Additional Time

The boat continues with the same acceleration for another \(5.0 \, \mathrm{s}\). We are given that the acceleration \(a = 0.8 \, \mathrm{m/s^2}\) and the time \(t = 5.0 \, \mathrm{s}\).

Step 4: Calculate New Speed After Additional Time

Use the speed-time relation formula \(v = u + at\) to find the new speed. The initial speed for this interval is \(8.0 \, \mathrm{m/s}\) (the speed after the first \(10 \, \mathrm{s}\)). Calculate: \(v = 8.0 + 0.8 \times 5 = 8.0 + 4.0 = 12.0 \, \mathrm{m/s}\).

Key Concepts

Average Acceleration

Average acceleration is a measure of how quickly an object changes its speed over a given period of time. It is a fundamental concept in kinematics and is calculated using the formula:
\[ a = \frac{v - u}{t} \]where \( a \) is the average acceleration, \( v \) is the final velocity, \( u \) is the initial velocity, and \( t \) is the time taken for this change.
When an object starts from rest, like the motorboat in our problem, the initial velocity \( u \) is 0. This simplifies our calculation since we only need to focus on the final velocity and the time interval.
To solve for average acceleration:

• Identify the initial and final velocities and the time period over which this change occurs.
• Substitute these values into the formula and solve for \( a \).

In the exercise, the boat's final speed is \(8.0\, \mathrm{m/s}\) achieved in \(10\, \mathrm{s}\), leading to an average acceleration of \(0.8\, \mathrm{m/s^2}\).

Speed-Time Relation

The speed-time relation is a key concept that helps us understand how an object moves with uniform acceleration. The fundamental formula is given by:\[ v = u + at \]
Here, \( v \) represents the final velocity of the object, \( u \) is its initial velocity, \( a \) is the constant acceleration, and \( t \) is the time duration.
In many situations, like the motorboat example, knowing this equation lets us extend our understanding of motion beyond a single time snapshot:

• If you know any three of the variables, you can solve for the fourth.
• This relation makes it easy to predict future velocities if the acceleration remains constant.

In our example, we initially used the speed-time relation to find the new speed of the boat after an additional \(5.0\, \mathrm{s}\) with its known acceleration.

Initial Velocity

Initial velocity is the speed at which an object begins its motion. This value is crucial when describing motion using the equations of kinematics. In physics, particularly in kinematics problems, starting conditions define how future motion is analyzed.
In our exercise, the motorboat starts from rest, meaning the initial velocity \( u \) of the boat is \(0\, \mathrm{m/s}\). This simplifies many calculations because:

• The initial kinetic energy is zero.
• The effect of acceleration becomes more straightforward to analyze over time.

Understanding the concept of initial velocity is essential for accurately predicting how changes in speed will unfold over time.

Final Velocity

Final velocity is the speed an object reaches after a given acceleration over a specific time period. It is the result of initial conditions and the effect of acceleration acting over a measured interval. The final velocity can be calculated through the speed-time relation:
\[ v = u + at \]
In kinematic problems, calculating the final velocity lets us know the object's speed at any given moment, assuming constant acceleration.
In our scenario:

• The boat first reaches a final velocity of \(8.0\, \mathrm{m/s}\) after \(10\, \mathrm{s}\).
• When the motion continues with the same average acceleration of \(0.8\, \mathrm{m/s^2}\) for another \(5.0\, \mathrm{s}\), the final velocity becomes \(12.0\, \mathrm{m/s}\).

Understanding how to calculate final velocity allows us to predict motion accurately for objects experiencing constant acceleration over time.