I'm having some trouble with this question
A 4200-kg rocket is traveling in outer space with a velocity of 120m/s toward the Sun. It needs to alter its course by 23 degrees, which can be done by shooting is rockets briefly in a direction perpendicular to it’s original motion. If the rocket gases are expelled at a speed of 2200 m/s relative to the rocket, what mass of gas must be expelled
I see that I need to find the other velocity
Vy = 120 tan 23
this gave me about 50.94 m/s
I do not know were to go from here
YOU ANSWER IS ABSOLUTELY INCORRECT I AM DOING (2200M=(4200-M)V COS 23( BY APPLYIND CONSERVATION OF MOMENTUM ) NOW WE GET( V=200M/S) SO REQUIRED ANS IS (V=200M/S)
To solve this problem and find the mass of gas that must be expelled, you can use the principle of conservation of momentum. The initial momentum of the rocket before shooting the gas is equal to the final momentum of the rocket and the expelled gas combined.
Step 1: Calculate the initial momentum of the rocket before shooting the gas.
Initial momentum (p_initial) = mass (m) × velocity (v)
= 4200 kg × 120 m/s
= 504,000 kg·m/s
Step 2: Calculate the final momentum of the rocket and the expelled gas.
Final momentum (p_final) = (m + x) × v_final
Here, x is the mass of the gas expelled and v_final is the final velocity.
Step 3: Determine the final velocity of the rocket and the expelled gas. Since the gas is expelled perpendicular to the original motion, the final velocity can be calculated using Pythagorean theorem.
v_final = sqrt(v^2 + Vy^2)
where v is the initial velocity of the rocket and Vy is the velocity gained perpendicular to the original motion.
v_final = sqrt((120 m/s)^2 + (50.94 m/s)^2)
= sqrt(14400 m^2/s^2 + 2594.5236 m^2/s^2)
= sqrt(16994.5236 m^2/s^2)
≈ 130.35 m/s
Step 4: Substitute the values into the momentum conservation equation.
p_initial = p_final
m × v = (m + x) × v_final
4200 kg × 120 m/s = (4200 kg + x) × 130.35 m/s
Step 5: Solve for x (mass of the gas).
Multiply out the equation and isolate x on one side:
4200 kg × 120 m/s = 130.35 m/s × (4200 kg + x)
504,000 kg·m/s = 130.35 m/s × 4200 kg + 130.35 m/s × x
504,000 kg·m/s = 546,315 kg·m/s + 130.35 m/s × x
504,000 kg·m/s - 546,315 kg·m/s = 130.35 m/s × x
- 42,315 kg·m/s = 130.35 m/s × x
-42,315 kg·m/s / 130.35 m/s = x
Step 6: Calculate x (mass of the gas).
x ≈ -324.15 kg
Since mass cannot be negative, we ignore the negative sign and take the absolute value.
Therefore, the mass of the gas that must be expelled is approximately 324.15 kg.
To find the mass of gas that must be expelled, you can use the principle of conservation of momentum.
First, calculate the initial momentum of the rocket in the y-direction (perpendicular to its original motion) using the formula:
initial momentum = mass of the rocket * velocity of the rocket in the y-direction
Since the rocket is initially moving only in the x-direction, the y-component of its velocity is given by Vy = 120 tan 23, which you have already calculated to be about 50.94 m/s. The mass of the rocket is 4200 kg.
So, initial momentum = 4200 kg * 50.94 m/s
Next, consider the rockets' change in momentum after the gas is expelled. The change in momentum is equal to the momentum of the emitted gas.
The velocity of the emitted gas relative to the rocket is given as 2200 m/s. However, we need to consider its velocity relative to the ground (sun) frame of reference.
Using vector subtraction, the velocity of the emitted gas relative to the ground is:
Vg = sqrt[(velocity of the rocket in x-direction)^2 + (velocity of the rocket in y-direction)^2] + velocity of the emitted gas relative to the rocket
The velocity of the rocket in the x-direction (Vx) is the given velocity of 120 m/s. The velocity of the rocket in the y-direction (Vy) is the value you calculated earlier, which is about 50.94 m/s.
So, Vg = sqrt[(120 m/s)^2 + (50.94 m/s)^2] + 2200 m/s
Calculate Vg using the equation above.
Next, calculate the momentum of the emitted gas relative to the ground frame of reference:
momentum of emitted gas = (mass of gas expelled) * (velocity of the emitted gas relative to the ground)
However, we need to find the mass of gas expelled. We can use the conservation of momentum equation:
initial momentum = final momentum
Substituting the values we have, we get:
4200 kg * 50.94 m/s = (mass of gas expelled) * (Vg)
From here, you can rearrange this equation to solve for the mass of the gas expelled:
mass of gas expelled = (4200 kg * 50.94 m/s) / Vg
Let the rocket be traveling in the x direction, and the boosters fire in the y direction.
Actually you need to alter the velocity so that the 120m/s is the x component, and a y component exists making the angle 23 degrees.
tan(theta) = y/x
tan(23) = y/120
Now you can use that to find the required change in momentum. By conservation of momentum, the momentum of the expelled gases equals the change in momentum of the rocket. Use that to find the mass of the expelled gas.