A particle moves with a velocity `(5hati-3hatj+6hatk) ms ^(-1)` under the influence of a constant force `vecF =(10 hati-10 hatj+20hatk)` N . The instantaneous power applied to the particle is

Options

A. 140 W
B. 40 W
C. 200 W
D. 170 W

Answer

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

### Step 1: Write down the formula for instantaneous power.
The instantaneous power \( P \) is given by the dot product of the force vector \( \vec{F} \) and the velocity vector \( \vec{v} \):
\[ P = \vec{F} \cdot \vec{v} \]

### Step 2: Identify the force and velocity vectors.
From the question, we have:
- Force vector \( \vec{F} = 10 \hat{i} - 10 \hat{j} + 20 \hat{k} \) N
- Velocity vector \( \vec{v} = 5 \hat{i} - 3 \hat{j} + 6 \hat{k} \) m/s

### Step 3: Calculate the dot product \( \vec{F} \cdot \vec{v} \).
The dot product is calculated as follows:
\[ \vec{F} \cdot \vec{v} = (10 \hat{i} - 10 \hat{j} + 20 \hat{k}) \cdot (5 \hat{i} - 3 \hat{j} + 6 \hat{k}) \]
Using the formula for the dot product:
\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]
we can substitute the components:
\[ = (10 \cdot 5) + (-10 \cdot -3) + (20 \cdot 6) \]

### Step 4: Perform the multiplication.
Calculating each term:
- First term: \( 10 \cdot 5 = 50 \)
- Second term: \( -10 \cdot -3 = 30 \)
- Third term: \( 20 \cdot 6 = 120 \)

### Step 5: Sum the results.
Now, add all the results together:
\[ P = 50 + 30 + 120 = 200 \text{ Watts} \]

### Final Answer:
The instantaneous power applied to the particle is \( 200 \text{ Watts} \).

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