Numerical/problem-solving

1. TITLE

How to solve Instantaneous Power using the Dot Product in NEET — Step-by-step

2. PROBLEM TYPE INTRO

This category of numerical tests your understanding of instantaneous power using vectors. We will need the formula connecting power, force, and velocity ($P = \mathbf{F} \cdot \mathbf{v}$). The most typical mistake students make is improperly multiplying the unit vectors ($\hat{i}, \hat{j}, \hat{k}$) or making sign errors when adding the products of the components.

3. MCQ

Question:
A particle moves with a velocity $\mathbf{v} = (5\hat{i} + 4\hat{j} + 3\hat{k})$ m/s under the influence of a constant force $\mathbf{F} = (3\hat{i} + 4\hat{j} - 5\hat{k})$ N. What is the instantaneous power applied to the particle?

A) 16 W
B) 46 W
C) 31 W
D) 50 W

4. STEP-BY-STEP SOLUTION

SOLUTION

Given:
Force, $\mathbf{F} = (3\hat{i} + 4\hat{j} - 5\hat{k})$ N
Velocity, $\mathbf{v} = (5\hat{i} + 4\hat{j} + 3\hat{k})$ m/s

To Find:
Instantaneous Power ($P$)

Formula:
$P = \mathbf{F} \cdot \mathbf{v}$
Use this formula when instantaneous force and velocity are given in vector components, as instantaneous power is defined as the dot product of force and instantaneous velocity.

Step 1: Write down the mathematical rule for the scalar (dot) product using components.
$P = F_x v_x + F_y v_y + F_z v_z$

Step 2: Multiply the corresponding magnitudes of the $\hat{i}$, $\hat{j}$, and $\hat{k}$ unit vectors. Remember that the dot product of perpendicular vectors (like $\hat{i} \cdot \hat{j}$) is zero.
$P = (3 \times 5) + (4 \times 4) + (-5 \times 3)$

Step 3: Calculate the final scalar value by being careful with the signs.
$P = 15 + 16 - 15$
$P = 16$

Answer: 16 W

5. WHERE STUDENTS GO WRONG

  • ❌ Mistake 1: Ignoring the negative sign in the components (Selecting Option B: 46 W by incorrectly adding $15 + 16 + 15$).
  • ❌ Mistake 2: Multiplying the scalar magnitudes of the vectors (Selecting Option D: 50 W, because $F = \sqrt{50}$ and $v = \sqrt{50}$) instead of taking the vector dot product.
  • ❌ Mistake 3: Forgetting that power is a scalar quantity and leaving the final answer with $\hat{i}, \hat{j}, \hat{k}$ direction components instead of calculating a single numerical value.

6. QUICK FORMULA REFERENCE

Formulas Used:
* $P = \mathbf{F} \cdot \mathbf{v}$ — Use when the force and velocity of an object are given in vector form and you need to calculate instantaneous power.
* $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$ — Use when calculating the scalar product of any two vectors given in rectangular components.

7. SIMILAR PROBLEM HINT

"If NEET changes the values, the method stays the same. Practice with: Force and position/displacement vectors given alongside a time $t$, requiring you to first find velocity by differentiating the position vector ($\mathbf{v} = d\mathbf{r}/dt$), and then applying the dot product trick."

8. NCERT REFERENCE

Based on: NCERT Class 11, Chapter 5, Example 5.1

9. CTA

More solved numericals → Work, Energy and Power Complete Guide
Practice problems → [Topic Test: https://knowmyexam.in/mock-tests]