Question

The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.75 x 104 kg/s, with an exhaust speed of 2.40 x 103 m/s. Find the acceleration (in m/s2) of the vehicle just as it lifted off the launch pad on the Earth, if the vehicle's initial mass was 3.00 x 106 kg. (Note: You must include the force of gravity to solve the problem.)

Answer 1

Answer:

$a=4.19 m/s^{2}$

Explanation:

In order to solve this problem, we must first draw a free body diagram of the situation. (See attached picture)

from this diagram, we can do the following sum of forces:

$\sum F=ma$

where m is the mass of the rocket and a is its acceleration.

$F_{e}-mg=ma$

where $F_{e}$ is the force of exhaust. We can solve this for the acceleration, so we get:

$a=\frac{F_{e}-mg}{m}$

we can find the force of exhaust by using the momentum formula:

Ft=mv

so we can solve this for the force:

$F_{e}=\frac{mv}{t}$

the problem already tells us what m/t is equal to, so we can directly substitute:

$F_{e}=(1.75x10^{4}kg/s)(2.40x10^3m/s)$

which yields:

$F_{e}=42x10^6 N$

So we can now substitute:

$a=\frac{42x10^6N-(3x10^6kg)(9.81m/s^{2})}{3x10^6kg}$

so:

$a=4.19 m/s^{2}$