Trajectory and parametric equation

In summary, for a particle following the parametric equations x(t)= 2t^2,y(t)= t^2-4t, z(t)= 3t-5, its velocity and acceleration can be found by taking the first and second derivatives of the position vector. To find the components of the velocity and acceleration in the direction given by i-3j+2k, the dot product of the velocity and acceleration vectors with this direction vector must be divided by the magnitude of the direction vector.
  • #1
crystalplane
11
0

Homework Statement


a particle is on a trajectory defined by the parametric equations x(t)= 2t^2,y(t)= t^2-4t,
z(t)= 3t-5, where t is the time.Find the components of its velocity and acceleration at time =1, in the direcion i-3j+2k.


Homework Equations


what i thought is r(t)=x(t) i +y(t) j +z(t) k, then take first derivative and get velocity vector. take the second derivative and get the acceleration vector.
However, the question provides a direction vector and i have no idea how to deal with the direction vector.

The Attempt at a Solution


r(t)'=4t i + (2t-4) j+(3) k.
r(t)''= (4) i + (2)j +(0) k
 
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  • #2
Find the direction cosines between velocity and acceleration vectors and given direction vector.
 
  • #3
You are on the correct path to solving the problem. Once you find r'(1) and r''(1) you will project the components of these two vectors on to the vector given by i-3j+2k. The dot product gives the component of one vector onto another. Since this component is multiplied by the magnitude of other vector, you must divide by the magnitude of the vector i.e.

Let A = i-3j+2k

then

[tex]\mbox{component}=\frac{r'(1)\bullet\mbox{A}}{\mid\mbox{A}\mid}[/tex]

Use the same approach for r''(1).
 

Related to Trajectory and parametric equation

1. What is the difference between trajectory and parametric equation?

Trajectory refers to the path followed by an object in motion, whereas parametric equation is a mathematical representation of this path using variables and parameters.

2. How are trajectory and parametric equation related?

The trajectory of an object can be described using a parametric equation, which allows us to calculate its position at any given time. Parametric equations also provide information about the speed and direction of the object's motion.

3. What are the variables used in a parametric equation for trajectory?

The variables used in a parametric equation for trajectory are time (t), position (x and y coordinates), velocity (v), and acceleration (a).

4. How are parametric equations used in real-world scenarios?

Parametric equations are commonly used in physics and engineering to describe the motion of objects, such as projectiles and satellites. They are also used in computer graphics to create animations and simulations.

5. Can parametric equations be used to calculate the trajectory of non-linear motion?

Yes, parametric equations can be used to describe the trajectory of any type of motion, including non-linear motion. In this case, the equations may involve trigonometric functions or higher order equations to account for the curvature of the path.

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