91Ó°ÊÓ

Newton's Second Law of Motion is crucial in understanding how forces act on objects. It tells us that the force acting on an object is the product of its mass and acceleration: \( F = ma \). This means:- If you know the mass and acceleration of an object, you can calculate the force.- Conversely, if you know the force and mass, the acceleration can be deduced.In the context of the tugboat exercise, the mass of the tugboat is large \((50 \mathrm{Mg} \equiv 50,000,000 \mathrm{~kg})\), and it accelerates to a speed of 6.944 meters per second over 35 seconds. By applying Newton's Second Law, you multiply the mass by the calculated acceleration (0.198 \( \mathrm{m/s^2} \)) to determine the propulsion force required for this motion, shown as \( F = 9.9 \times 10^6 \mathrm{~N} \) (Newtons).Understanding this principle explains why engines in large ships must exert enormous forces to achieve seemingly modest changes in speed.

Kinematic equations help us describe the motion of objects without considering the forces that cause the motion. They are derived under the assumption that acceleration is constant.A key kinematic equation used in this exercise is: \[v = u + at\]where:

• \( v \) is the final velocity,
• \( u \) is the initial velocity (which is 0 in this exercise),
• \( a \) is the acceleration,
• \( t \) is the time taken.

Starting from rest (\( u = 0 \)), the tugboat reaches a velocity of 6.944 \( \mathrm{m/s} \) in 35 seconds, as calculated from the given velocity in the problem. The acceleration \( a \) is then found using this formula: \( a = \frac{v}{t} \). This allows us to break down complex motion into manageable steps, where just by knowing two factors, such as time and final velocity, we can calculate the third, the acceleration.

In this situation, propulsion force is the force exerted by the tugboat's engines to generate movement. It's the primary force required to accelerate the tugboat from rest.Here's a step-by-step breakdown:1. **Determine Mass**: The tugboat's mass is given as 50 Megagrams (Mg), which translates to \( 50,000,000 \mathrm{~kg}\).2. **Find Acceleration**: Using the kinematic equations, the acceleration was found to be 0.198 \( \mathrm{m/s^2} \).3. **Calculate Force**: Applying Newton's Second Law again, \( F = ma \), the propulsion force is:\[F = 50,000,000 \times 0.198 = 9.9 \times 10^6 \mathrm{~N}\]Understanding how to effectively compute propulsion force is crucial in maritime engineering. It ensures that engines and machinery are adequately rated for their required duties. This means matching operational needs to engineering capabilities to ensure smooth, safe, and efficient transport across water.