Finding the Point to Shut Off Engines for a Traveler in Space

In summary: The only other option I can think of is graphing it, and with sketching skill that would be the hard way!
  • #1
Physics1
21
0

Homework Statement



A traveler in space is moving left and right on y = x^2. He shuts off the engines and continues on the tangent line until he reaches point (4, 15). At what point on the curve should he shut off the engines to reach that point?

The Attempt at a Solution



Ok, the derivative is 2x. I know have to solve for the slope of the tangent line.

Rise over run = (15-a^2) / (4 - a)

(a, a^2) is the point to shut off. I get a^2 - 8a + 15 as the quadratic equation but the value 4 from it isn't correct when I plug it into 2x. What am I doing wrong?
 
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  • #2
What is the general form of a straight line?
 
  • #3
Hootenanny said:
What is the general form of a straight line?

y = mx + b

I know. I did that but it isn't the correct answer when looking at the graph.

I got y = 8x - 17. It doesn't touch the graph
 
  • #4
So we now have to equations for the gradient and our point x=a;

[tex]m = 2a \hspace{1cm};\hspace{1cm}m = \frac{15-a^2}{4-a}[/tex]

Can you now solve for a?
 
  • #5
Hootenanny said:
So we now have to equations for the gradient and our point x=a;

[tex]m = 2a \hspace{1cm};\hspace{1cm}m = \frac{15-a^2}{4-a}[/tex]

Can you now solve for a?

I see what I did wrong now. I did the quadratic wrong because I accidentally put a wrong number in so I had to do the quadratic equation which led to a wrong number. 3 works for a and the point is (3, 9). Isn't there another way to solve this?
 
  • #6
Physics1 said:
I see what I did wrong now. I did the quadratic wrong because I accidentally put a wrong number in so I had to do the quadratic equation which led to a wrong number. 3 works for a and the point is (3, 9). Isn't there another way to solve this?
Not that I can think of...
 
  • #7
Hootenanny said:
Not that I can think of...

My teacher said there's an easy way (I'm assuming it's this way) and a hard way.
 
  • #8
Physics1 said:
My teacher said there's an easy way (I'm assuming it's this way) and a hard way.
The only other option I can think of is graphing it, and with sketching skill that would be the hard way!
 

What is the point at which the engines should be shut off for a traveler in space?

The point at which the engines should be shut off for a traveler in space depends on a variety of factors, including the destination, the spacecraft's velocity, and the desired orbit. However, in general, the engines should be shut off when the desired velocity and trajectory have been achieved.

How is the point for shutting off engines determined?

The point for shutting off engines is determined by calculating the required velocity and trajectory for the spacecraft to reach its destination, taking into account factors such as gravitational pull and atmospheric drag. This information is then used to determine the optimal point at which the engines should be shut off.

What happens if the engines are shut off too early or too late?

If the engines are shut off too early, the spacecraft may not reach its intended destination or may not have enough velocity to enter the desired orbit. On the other hand, if the engines are shut off too late, the spacecraft may overshoot its destination or enter an undesired orbit. In both cases, the mission may fail or require additional fuel and resources to correct the trajectory.

How does the weight of the spacecraft affect the point for shutting off engines?

The weight of the spacecraft does affect the point for shutting off engines, as it directly impacts the amount of fuel needed to achieve the desired velocity and trajectory. A heavier spacecraft will require more fuel and may need to shut off its engines at a different point than a lighter spacecraft.

Are there any safety considerations when determining the point for shutting off engines?

Yes, safety considerations are always taken into account when determining the point for shutting off engines. This includes ensuring that the spacecraft has enough fuel for a safe return journey, as well as avoiding potential collisions with other objects in space. In addition, the spacecraft's trajectory must be carefully planned to avoid any potential hazards or obstacles.

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