Question

A motor boat is moving in a river with velocity $\vec{v} = 7\hat{i} + 2\hat{j} - 5\hat{k}$ kms$^{-1}$. If the flow water offers resistive force $\vec{F} = 9\hat{i} + 3\hat{j} - 3\hat{k}$ N, then the power of the boat is

• A. 13 W
• B. 69 W
• C. 12 W
• D. 84 W

Hint:

Power is the dot product of force and velocity. Ensure that the units of force and velocity are consistent (preferably SI units) to obtain the power in Watts. If the units are not consistent, appropriate conversions must be made.

Answer 1

The Correct Option is D

Step 1: Identify the given velocity and force vectors.
Velocity vector: $\vec{v} = 7\hat{i} + 2\hat{j} - 5\hat{k}$ kms$^{-1}$ Resistive force vector: $\vec{F} = 9\hat{i} + 3\hat{j} - 3\hat{k}$ N
Step 2: Recall the formula for power.
The power $P$ is given by the dot product of the force and the velocity: $P = \vec{F} \cdot \vec{v}$.
Step 3: Perform the dot product.
$P = (9\hat{i} + 3\hat{j} - 3\hat{k}) \cdot (7\hat{i} + 2\hat{j} - 5\hat{k})$ $P = (9)(7) + (3)(2) + (-3)(-5)$ $P = 63 + 6 + 15$ $P = 84$
Step 4: Consider the units.
The unit of force is Newton (N) and the unit of velocity is kms$^{-1}$. The unit of power would be N kms$^{-1}$. $1 \text{ N kms}^{-1} = (1 \text{ kg m s}^{-2}) (1000 \text{ m s}^{-1}) = 1000 \text{ kg m}^2 \text{ s}^{-3} = 1000 \text{ W} = 1 \text{ kW}$. So, the power is 84 kW. However, if we assume there was a typo and the velocity was meant to be in m s$^{-1}$: $\vec{v} = 7\hat{i} + 2\hat{j} - 5\hat{k}$ ms$^{-1}$ $P = (9)(7) + (3)(2) + (-3)(-5) = 63 + 6 + 15 = 84$ W. Given the options, it is highly likely that the velocity unit should have been m s$^{-1}$. Assuming the velocity is in m s$^{-1}$, the power of the boat is $\boxed{84 \text{ W}}$.