A particle moves with the velocity `vecv=(5hati+2hatj-hatk)ms^(-1)` under the influence of a constant force, `vecF=(2hati+5hatj-10hatk)N.` The instantaneous power applied is

Options

A. 5 W
B. 10 W
C. 20 W
D. 30 W

Answer

To find the instantaneous power applied to a particle moving under the influence of a constant force, we can use the formula for power in terms of force and velocity:

**Step 1: Write down the given vectors.**
- The velocity vector is given as:
\[ \vec{v} = 5\hat{i} + 2\hat{j} - \hat{k} \, \text{m/s} \]
- The force vector is given as:
\[ \vec{F} = 2\hat{i} + 5\hat{j} - 10\hat{k} \, \text{N} \]

**Step 2: Use the formula for instantaneous power.**
The instantaneous power \( P \) can be calculated using the dot product of the force and velocity vectors:
\[ P = \vec{F} \cdot \vec{v} \]

**Step 3: Calculate the dot product.**
To compute the dot product, we multiply the corresponding components of the vectors and then sum them up:
\[ P = (2\hat{i} + 5\hat{j} - 10\hat{k}) \cdot (5\hat{i} + 2\hat{j} - \hat{k}) \]
Calculating the dot product:
\[ P = (2 \times 5) + (5 \times 2) + (-10 \times -1) \]
\[ P = 10 + 10 + 10 \]
\[ P = 30 \, \text{W} \]

**Step 4: State the final answer.**
Thus, the instantaneous power applied is:
\[ P = 30 \, \text{W} \]

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