Numerical/Problem-solving

1. TITLE

How to solve rocket hovering problems in NEET — Step-by-step

2. PROBLEM TYPE INTRO

This is a "variable mass" dynamics problem under Newton's Second Law. We need the thrust formula ($F = v \frac{dm}{dt}$) and the weight formula ($W = mg$). A typical mistake students make is equating thrust directly to the rocket's mass instead of its weight, or forgetting to convert exhaust velocity units.

3. MCQ

A rocket with a mass of $20,000\text{ kg}$ needs to hover stationary above the ground. If it ejects gas at an exhaust velocity of $1.0\text{ km/s}$, what is the rate of fuel combustion required to keep the rocket hovering? (Take $g = 10\text{ m/s}^2$).

(a) $20\text{ kg/s}$
(b) $200\text{ kg/s}$
(c) $200,000\text{ kg/s}$
(d) $2 \times 10^8\text{ kg/s}$

4. STEP-BY-STEP SOLUTION

SOLUTION

Given:
* Mass of rocket ($m$) = $20,000\text{ kg}$
* Exhaust velocity ($v$) = $1.0\text{ km/s} = 1000\text{ m/s}$
* Acceleration due to gravity ($g$) = $10\text{ m/s}^2$

To Find:
* Fuel combustion rate ($\frac{dm}{dt}$) in $\text{kg/s}$

Formula:
* Thrust Force = Weight of Rocket
* $v \frac{dm}{dt} = mg$
* Explain: Use this formula when a variable mass system (like a rocket) is in equilibrium (hovering). According to Newton's First Law, if a body is stationary, the net external force on it is zero. Therefore, the upward thrust generated by the rate of change of momentum must perfectly balance the downward gravitational force.

Step 1: Calculate the total weight of the rocket
$W = mg = 20,000\text{ kg} \times 10\text{ m/s}^2 = 200,000\text{ N}$

Step 2: Set up the thrust equation using Newton's Second Law
$F = v \frac{dm}{dt}$
$200,000\text{ N} = 1000\text{ m/s} \times \frac{dm}{dt}$

Step 3: Solve for the rate of fuel combustion
$\frac{dm}{dt} = \frac{200,000}{1000} = 200$

Answer: (b) $200\text{ kg/s}$

5. WHERE STUDENTS GO WRONG

  • ❌ Mistake 1: Forgetting to convert the exhaust velocity from $\text{km/s}$ to $\text{m/s}$. Plugging in $v = 1$ leads to the massive $200,000\text{ kg/s}$ error (Option C).
  • ❌ Mistake 2: Equating thrust directly to mass instead of weight (forgetting to multiply by $g$). Setting $1000 \times \frac{dm}{dt} = 20,000$ leads to $20\text{ kg/s}$ (Option A).
  • ❌ Mistake 3: Multiplying weight and exhaust velocity instead of dividing them, using an incorrect arrangement of the formula (Option D).

6. QUICK FORMULA REFERENCE

Formulas Used:
* $F = v \frac{dm}{dt}$ — Use when calculating the thrust force generated by ejecting mass at a constant relative velocity.
* $v \frac{dm}{dt} = mg$ — Use when a rocket is hovering or stationary (acceleration is zero, meaning net force is balanced).

7. SIMILAR PROBLEM HINT

"If NEET changes the values, the method stays the same. Practice with: calculating the initial upward acceleration if the fuel combustion rate is given as slightly greater than the hovering rate."

8. NCERT REFERENCE

Based on: NCERT Class 11, Chapter 4, Exercise 4.9.

9. CTA

More solved numericals → Laws of Motion Guide
Practice problems →Mock exams