Video Transcript
A rocket was ascending vertically,
projecting its burnt fuel at 3,600 kilometers per hour vertically downwards. Given that, for every eight
seconds, it expelled three kilograms of fuel, find the elevator force generated by
the rocket’s engine.
Now, we’ve been asked to find the
force generated by the rocket’s engine. And we can also see that the
rocket’s mass is varying since it’s expelling fuel. Therefore, we will need to use
Newton’s second law for variable mass, which tells us that 𝐹 is equal to 𝑚d𝑣 by
d𝑡 plus 𝑣d𝑚 by d𝑡.
Now when we look a bit closer at
the question, we may notice that the information we’ve been given is in fact about
the fuel and not the rocket. We are told that it is projecting
it’s burnt fuel at 3,600 kilometers per hour and that every eight seconds three
kilograms of fuel is expelled. This means if we’re calculating
using these values, we will in fact be finding the force acting upon the fuel which
is expelled from the rocket. This force 𝐹 will be acting
vertically downwards.
Now, in order to find the elevator
force which propels the rocket vertically upwards, we can use Newton’s third law,
which tells us that this force will be equal and opposite to the force of the fuel
being projected vertically downwards. Hence, it will be a force with the
same magnitude 𝐹 just acting vertically upwards.
So, let’s use Newton’s second law
for variable mass to find the force acting on the fuel, which is being expelled
vertically downwards. We have that the velocity of the
fuel will be 3,600 kilometers per hour. However, since this is acting
vertically downwards, we can write this as negative 3,600 kilometers per hour. The units of this velocity are
kilometers per hour. However, since we’re going to be
wanting to work in kilograms, meters, seconds, and newtons, we need to convert this
velocity to meters per second. Since there are 1,000 meters in a
kilometer and 3,600 seconds in an hour, we need to multiply the negative 3,600 by
1,000 over 3,600. So, our velocity is negative 1,000
meters per second.
Since 𝑣 is a constant, when we
differentiate it to find d𝑣 by d𝑡, we will see that it is equal to zero. Hence, when we substitute it into
our formula, this will make the first term be equal to zero. Due to this constant velocity, we
can rewrite our formula for the force as 𝐹 is equal to 𝑣d𝑚 by d𝑡. We have just found 𝑣; therefore,
the only thing we need to find 𝐹 is d𝑚 by d𝑡. This is the rate of change of the
mass.
We’re told that for every eight
seconds the rocket expelled three kilograms of fuel. Since we are considering the force
acting on the fuel which is being expelled from the rocket, this statement tells us
that the mass of fuel which has been expelled is increasing by three kilograms every
eight seconds. To find the rate of fuel being
expelled every second, we simply need to divide the three kilograms by the eight
seconds. So, the rate of change of mass is
equal to three over eight kilograms per second.
We can now substitute this value
along with the velocity into our equation for the force. The force is therefore equal to
negative 1,000 multiplied by three over eight, which simplifies to negative 375
newtons. Here, we have nearly reached our
solution. However, this is the force acting
upon the fuel which has been expelled from the rocket. The elevator force generated by the
rocket’s engines will be equal and opposite to this force. Our solution is that the elevator
force is equal to 375 newtons.
