Math Problem Statement
A spacecraft moves along a path described by the parametric equations shown below for the first 200 s after launch. Here, x and y are measured in meters, and t is measured in seconds. Find the magnitude and direction of the velocity of the spacecraft 20.0 s and 200 s after launch. xequals50 left parenthesis StartRoot 1 plus t Superscript 4 EndRoot minus 1 right parenthesis, yequals20 t Superscript 3 divided by 2
Solution
Given the parametric equations for the spacecraft's path:
To find the magnitude and direction of the velocity at s and s, we follow these steps:
1. Find the velocity components.
The velocity components and are given by the derivatives of the position functions and with respect to time :
- Velocity in the x-direction:
Using the chain rule:
- Velocity in the y-direction:
2. Find the magnitude of the velocity vector.
The magnitude of the velocity is given by the Pythagorean theorem:
3. Find the direction of the velocity.
The direction of the velocity (angle with respect to the x-axis) is:
At s:
-
Find :
-
Find :
-
Find the magnitude of velocity at :
-
Find the direction at :
At s:
-
Find :
-
Find :
-
Find the magnitude of velocity at :
-
Find the direction at :
Summary of Results:
-
At s:
- Magnitude of velocity:
- Direction:
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At s:
- Magnitude of velocity:
- Direction:
Do you want more details or clarifications on any part of the process?
Here are 5 related questions you might find interesting:
- How do we calculate acceleration from parametric equations?
- Can we find the displacement of the spacecraft after 200 seconds?
- How do velocity and acceleration differ in parametric motion?
- How would these results change if the functions had different forms (e.g., quadratic or linear)?
- What is the role of time in determining the curvature of the spacecraft's path?
Tip: The derivative of a parametric equation gives the rate of change of the respective variable, making it a useful tool for finding velocity and acceleration.
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Math Problem Analysis
Mathematical Concepts
Calculus
Parametric Equations
Velocity
Trigonometry
Formulas
Velocity components: v_x(t) = dx/dt, v_y(t) = dy/dt
Magnitude of velocity: v(t) = √(v_x^2 + v_y^2)
Direction of velocity: θ(t) = tan⁻¹(v_y/v_x)
Theorems
Pythagorean Theorem
Chain Rule for Derivatives
Suitable Grade Level
Advanced High School (Grades 11-12), College Level
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