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Difficulty: MediumMechanics and Motion

A 400 kg400\text{ kg} patrol boat accelerates uniformly from rest to a speed of 18 m/s18\text{ m/s} in 6 seconds6\text{ seconds} across calm water. If a constant hydrodynamic resistance force of 300 N300\text{ N} opposes the motion, what forward thrust force must the boat's engine produce?

  1. A
    1,200 N1,200\text{ N}
  2. 1,500 N1,500\text{ N}Answer
  3. C
    900 N900\text{ N}
  4. D
    4,200 N4,200\text{ N}

Answer

The boat's engine must produce a forward thrust of 1,500 N1,500\text{ N}.
To find the total thrust, first determine the acceleration from the velocity change (18 m/s÷6 s=3 m/s218\text{ m/s} \div 6\text{ s} = 3\text{ m/s}^2). Using Newton's Second Law (F=maF = ma), the net forward force required is 400 kg×3 m/s2=1,200 N400\text{ kg} \times 3\text{ m/s}^2 = 1,200\text{ N}. Because 300 N300\text{ N} of water resistance opposes the boat, the engine must supply enough forward force to overcome drag plus provide the net accelerating force: 1,200 N+300 N=1,500 N1,200\text{ N} + 300\text{ N} = 1,500\text{ N}.

Step-by-Step Solution

1
Calculate the acceleration of the patrol boat using kinematic relations.
a=vf−vit=18 m/s−0 m/s6 s=3 m/s2a = \frac{v_f - v_i}{t} = \frac{18\text{ m/s} - 0\text{ m/s}}{6\text{ s}} = 3\text{ m/s}^2
Uniform acceleration is defined as the change in velocity divided by the elapsed time.
2
Determine the net horizontal force required to accelerate the boat using Newton's Second Law.
Fnet=m⋅a=400 kg×3 m/s2=1,200 NF_{\text{net}} = m \cdot a = 400\text{ kg} \times 3\text{ m/s}^2 = 1,200\text{ N}
The net force acting on any mass is directly proportional to its acceleration (F=maF = ma).
3
Apply the net force equation along the horizontal axis to solve for engine thrust.
Fthrust−Fresistance=Fnet  ⟹  Fthrust=1,200 N+300 N=1,500 NF_{\text{thrust}} - F_{\text{resistance}} = F_{\text{net}} \implies F_{\text{thrust}} = 1,200\text{ N} + 300\text{ N} = 1,500\text{ N}
The engine thrust must overcome the backward resistive drag while providing the remaining force needed for forward acceleration.

Key Concept

Newton's Second Law and Net Force in One Dimension
Estimated Time:1m 0s
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Topics

Mechanics and Motion

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