Parametric equations for trajectory

In summary, the trajectory of the jet, displayed in the yz-plane as it would appear to an observer at the point (1, 0, 0), can be represented by the formula y(t) = 1 and z(t) = t. However, for the actual position of the jet in 3 dimensions, the formula would be (1- 5t, 1, t). To find the straight line from (1, 0, 0) passing through this trajectory, the parameter s can be used to represent (1- 5st, 1+ s, ts). This line intersects the yz-plane at (0, 1+1/5t, 1/5)
  • #1
plexus0208
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Homework Statement


A jet takes off from (1, 1, 0) at time t = 0 and moves with constant speed v = (−5, 0, 1).
In a flight simulator, the trajectory of the jet is displayed in the yz-plane as it would appear to an observer at the point (1, 0, 0). Find the formula (in the form y = y(t), z = z(t)) for the trajectory on the screen.

Homework Equations



The Attempt at a Solution


y(t) = 1
z(t) = t
Is that right?
 
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  • #2
That would be the "parallel projection" onto the yz-plane, not "it would appear to an observer at the point (1, 0, 0)". The position of the jet, in 3 dimensions, is (1- 5t, 1, t). A straight line from (1, 0, 0) through that, in parameter s, would be (1- 5st, 1+ s, ts). That passes through the "yz-plane" (x= 0) when 1- 5st= 0 or s= 1/5t: (0, 1+1/5t, 1/5).
 

1. What are parametric equations for trajectory?

Parametric equations for trajectory are mathematical equations that describe the motion of an object in terms of time. These equations include separate equations for the x and y coordinates, where the values of the coordinates are dependent on a parameter, typically time.

2. How are parametric equations for trajectory used?

Parametric equations for trajectory are used in physics and engineering to model the motion of objects. They can also be used in computer graphics to create animations of moving objects.

3. What are the advantages of using parametric equations for trajectory?

One advantage of using parametric equations for trajectory is that they can accurately model the motion of objects in three-dimensional space. They also allow for more complex trajectories, such as curves and spirals, to be described using simple equations.

4. Can parametric equations for trajectory be used for any type of motion?

Yes, parametric equations for trajectory can be used for any type of motion, including linear, circular, and projectile motion. As long as the motion can be described in terms of x and y coordinates, parametric equations can be used.

5. How do you convert parametric equations for trajectory into Cartesian equations?

To convert parametric equations for trajectory into Cartesian equations, the equations for the x and y coordinates can be solved for the parameter, typically time. The resulting equations can then be substituted into each other to eliminate the parameter and form a single equation in terms of x and y.

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