Calculating Final Speed & Angle of a Rocket Motion

In summary, the problem involves a rocket traveling at 20 km/s in a non-gravity space and turning on an engine to fix its direction of motion. The engine pushes the gases with a constant speed of 3 km/s perpendicular to the direction of motion. The engine is turned on until the mass of the rocket declines for 1/4 of the initial. To find the angle of change in direction and the final speed, the momentum conservation equation is used in the y-direction, and trigonometry is used to determine the angle. It is assumed that the gas is ejected perpendicularly to the initial direction of motion. The resulting angle is between 0 and 2.6 degrees, with a final velocity between 0
  • #1
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Homework Statement



A rocket is traveling with a speed 20 km/s in a non-gravity space. To fix the direction of motion, it turns on an engine, which pushes the gasses with constant speed 3 km/s w.r.t. the rocket perpendicularly to the direction of its motion. The engine is on till the mass of the rocket declines for 1/4 of the initial.

For what angle has the rocket changed the direction of motion and what is its final speed?


Homework Equations



x-axis: d( p(rocket) ) = - d( m(gasses) ) v(gasses) cos( alpha+d(alpha) )

dm/dt * v(rocket_final) + dv/dt m( rocket_new )= - dm/dt * v(gasses) (cos(alpha)cos(d(alpha)) - sin(alpha)sin(d(alpha)) )

dm/dt * v(rocket_final) + dv/dt ( m- dm/dt *t) = - dm/dt * v(gasses) *cos(alpha)cos(d(alpha))



The Attempt at a Solution



I tried to express the change in momentum of the rocket w.r.t. the angle of motion like above for both x- and y- axis, but I get unknown velocity and angle.
 
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  • #2
Taking the direction of motion of the rocket to be the x-axis, write the momentum conservation equation in the y direction. This will give you the final velocity of the rocket in the y direction.

Now, you know both the x and y components of the final velocity, and you can calculate the angle that the velocity makes with the x-axis using trigonometry. (Hint: Draw a picture. What is tan(θ) in terms of Vx and Vy?)
 
  • #3
Thank you, dx.

But the problem I see is, that the direction of gasses also changes with moving rocket. Therefore also Vx changes. The whole momentum does not go into Vy.
 
  • #4
The direction of motion changes, but the orientation of the rocket stays the same. So the direction in which the gas is ejected is the same.
 
  • #5
What refers orientation of the rocket to, dx?

I thought of this problem again and maybe this is a solution.

Let us say, that the rocket would push the gasses with a given Vr=3 km/s in front of itself. Then its final Vx would be 19,2 km/s.

On the other hand, if the rocket would push the gasses with a given Vr perpendicularly to the initial direction of motion, its final Vy would be 0,86 km/s.

Therefore in any case 0<Vy<0,86 and 19,2<Vx<20. Therefore the angle for which it turns is
0< theta < 2,6 degrees.


Using the fact about the angle, we can see, that the change in Vx in x-axis can be neglected, since sin(2,6)~0,045. Also since cos(2,6)~0,998, we can take V=Vr for the relative velocity of gasses in y-axis during turning.

Therefore we can really assume that the gasses are ejected perpendicularly to the initial direction of rocket's motion.
 

1. How do I calculate the final speed of a rocket in motion?

The final speed of a rocket can be calculated using the formula: Vf = Vi + at, where Vf is the final velocity, Vi is the initial velocity, a is the acceleration, and t is the time taken. It is important to note that the acceleration and time should be in the same units.

2. What is the formula for calculating the angle of a rocket's motion?

The angle of a rocket's motion can be calculated using the formula: θ = tan-1 (Vy/Vx), where θ is the angle, Vy is the vertical component of the velocity, and Vx is the horizontal component of the velocity. This formula is based on the trigonometric ratio of tangent.

3. How do I determine the initial velocity of a rocket?

The initial velocity of a rocket can be determined by dividing the change in distance by the change in time. In other words, Vi = (Vf - at)/t. It is important to ensure that the units for time and distance are consistent.

4. Can the final speed and angle of a rocket be calculated simultaneously?

Yes, the final speed and angle of a rocket can be calculated simultaneously by using the equation: Vf2 = Vi2 + 2ad, where Vf is the final velocity, Vi is the initial velocity, a is the acceleration, and d is the distance travelled. Once the final speed is determined, the angle can be calculated using the formula mentioned in question 2.

5. How can air resistance affect the calculation of final speed and angle of a rocket?

Air resistance can significantly affect the calculation of final speed and angle of a rocket as it can slow down the rocket's motion and change its direction. To account for air resistance, the formula for acceleration should be modified to include the force of air resistance, and the calculations should be adjusted accordingly.

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