Introduction to Rocket Propulsion
Rocket propulsion is a classic application of Newton's Third Law (action and reaction) and the principle of conservation of momentum. When a rocket expels gas out of its rear at high speed, it generates a force known as thrust, which pushes the rocket forward.
The Physics Concept
The thrust force ($F$) exerted on a rocket is equal to the rate at which momentum is expelled. Mathematically, this is expressed as: $$F = v_{rel} \cdot \frac{dm}{dt}$$ Where:
- $v_{rel}$ is the velocity of the gas relative to the rocket.
- $\frac{dm}{dt}$ is the rate at which the mass of the gas is emitted.
Once we find the thrust force, we use Newton's Second Law ($F = ma$) to find the acceleration ($a$): $$a = \frac{F}{m_{rocket}}$$
Step-by-Step Solution
Given data:
- Rate of mass emission ($\frac{dm}{dt}$) = $0.2 \text{ kg/s}$
- Relative velocity ($v_{rel}$) = $40 \text{ m/s}$
- Mass of rocket ($m$) = $4 \text{ kg}$
Step 1: Calculate the Thrust Force
$$F = v_{rel} \cdot \frac{dm}{dt}$$ $$F = 40 \text{ m/s} \cdot 0.2 \text{ kg/s} = 8 \text{ N}$$
Step 2: Calculate Initial Acceleration
Using $F = ma$, we rearrange for acceleration: $$a = \frac{F}{m}$$ $$a = \frac{8 \text{ N}}{4 \text{ kg}} = 2 \text{ m/s}^2$$
Result: The initial acceleration of the toy rocket is $2 \text{ m/s}^2$.