Analysis of Particle’s Motion
Step 1: Position Function
Comment: The position function describes the particle’s position at any given time t.
s(t) = t³ – 2t
Step 2: Velocity Function (Derivative of Position Function)
Comment: The velocity function represents the rate of change of the position with respect to time. It is the derivative of the position function.
v(t) = 3t² – 2
Step 3: Point where Velocity is Zero (Change in Direction)
Comment: To find when the particle changes direction, we set the velocity function to zero and solve for t.
3t² – 2 = 0
Solution: t = ±√(2/3)
Step 4: Evaluate Position Function at Key Points
Comment: We evaluate the position function at the start, the point of direction change, and the end to determine the particle’s positions at these times.
For t = 0: s(0) = 0
For t = √(2/3): s(√(2/3)) = -4√2/3
For t = 2: s(2) = 4
Step 5: Calculate Distance Using Absolute Values
Comment: We’ll compute the distance the particle traveled between the key points using the absolute difference in the position values.
Distance from t = 0 to t = √(2/3): |s(√(2/3)) – s(0)| = 4√2/3 ≈ 1.88562 units
Distance from t = √(2/3) to t = 2: |s(2) – s(√(2/3))| = 4 + 4√2/3 ≈ 5.88562 units
Step 6: Total Distance Traveled
Comment: The total distance traveled by the particle is the sum of the distances calculated in the previous step.
Total Distance = 1.88562 + 5.88562 = 7.77124 units
Approximated Total Distance: ≈ 7.77 units
Particle Motion Analysis
Position Function:
The particle’s position at any given time t is described by the function:
s(t) = t⁴ – 4t²
Velocity Function (Derivative of Position Function):
The velocity function represents the rate of change of the position with respect to time. It is the derivative of the position function.
v(t) = 4t³ – 8t
Direction Change Points:
To find when the particle changes direction, we set the velocity function to zero and solve for t:
4t³ – 8t = 0
This gives us three potential points of interest: t = 0, t = √2, and t = -√2.
Evaluate Position Function at Key Points:
- For t = 0: s(0) = 0
- For t = √2: s(√2) = 4 – 8 = -4
- For t = -√2: s(-√2) = 4 – 8 = -4
Calculate Distance Using Absolute Values:
1. Distance from t = -√2 to t = 0:
Distance = |s(0) – s(-√2)| = 4
2. Distance from t = 0 to t = √2:
Distance = |s(√2) – s(0)| = 4
The total distance traveled by the particle between t = -√2 and t = √2 is 8 units.
Graph:
Note: The graph shows the position function s(t) = t⁴ – 4t² over the interval [-1.5, 1.5]. The points of interest are t = -√2, t = 0, and t = √2.