A rocket of lift off mass of 20,000 kg is blasted with initial acceleration of $5~m/s^{2}$. Calculate the initial force of blast.

Question

Accepted Answer

The correct option is (4):

To calculate the initial force of the blast, we need to consider all forces acting on the rocket during lift-off. According to Newton's second law of motion, the net force ($F_{	ext{net}}$) acting on an object is equal to its mass ($m$) multiplied by its acceleration ($a$), i.e., $F_{	ext{net}} = ma$. When a rocket blasts off, two primary forces are acting on it: the upward thrust (initial blast force, $F_{	ext{blast}}$) and the downward force due to gravity (weight, $F_g$). Since the rocket is accelerating upwards, the net force is in the upward direction, meaning the blast force is greater than the gravitational force.

Therefore, the net force can be expressed as:
$F\_{	ext{net}} = F\_{	ext{blast}} - F\_g$

We know that $F_{	ext{net}} = ma$ and the force due to gravity is $F_g = mg$, where $g$ is the acceleration due to gravity, approximately $9.8 	ext{ m/s}^2$.

Substituting these into the equation:
$ma = F\_{	ext{blast}} - mg$

Rearranging to solve for $F\_{	ext{blast}}$:
$F\_{	ext{blast}} = ma + mg = m(a + g)$

Given:
Mass of the rocket, $m = 20,000 	ext{ kg}$
Initial acceleration, $a = 5 	ext{ m/s}^2$
Acceleration due to gravity, $g = 9.8 	ext{ m/s}^2$

Now, let's calculate the initial force of the blast:
$F\_{	ext{blast}} = 20,000 	ext{ kg} 	imes (5 	ext{ m/s}^2 + 9.8 	ext{ m/s}^2)$
$F\_{	ext{blast}} = 20,000 	ext{ kg} 	imes (14.8 	ext{ m/s}^2)$
$F\_{	ext{blast}} = 296,000 	ext{ N}$
This can be written in scientific notation as $2.96 	imes 10^5 	ext{ N}$.

The incorrect option is (1): $10^8 	ext{ N}$. This value is significantly higher than the calculated force, suggesting a miscalculation or an incorrect understanding of the forces involved.

The incorrect option is (2): $4 	imes 10^2 	ext{ N}$. This value is much too small for the blast force of a rocket with such a large mass and acceleration, likely representing an error in magnitude calculation.

The incorrect option is (3): $5 	imes 10^6 	ext{ N}$. This value is also considerably higher than the correct blast force, indicating a potential error in applying Newton's second law or in the numerical computation.