Math Problem Statement
Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. a(t) = 17t i + et j + e−t k, v(0) = k, r(0) = j + k
Solution
To find the position vector of the particle, we need to integrate the given acceleration vector twice and apply the initial conditions for velocity and position.
Step 1: Find the velocity vector
The acceleration vector is given as: We integrate to find the velocity vector .
Velocity in the -direction:
Velocity in the -direction:
Velocity in the -direction:
Thus, the general form of the velocity vector is:
Step 2: Apply the initial condition
From , we know:
This gives the following conditions:
Thus, the velocity vector becomes:
Step 3: Find the position vector
We now integrate to find .
Position in the -direction:
Position in the -direction:
Position in the -direction:
Thus, the general form of the position vector is:
Step 4: Apply the initial condition
From , we know:
This gives the following conditions:
Thus, the position vector becomes:
Final Answer:
The position vector of the particle is:
Would you like further details on any part of the solution or have any questions?
Here are 5 related questions:
- How do you find velocity from a given acceleration vector?
- What is the relationship between position, velocity, and acceleration vectors?
- Can you derive the position vector if given initial conditions for position only?
- How does the exponential function behave in differential equations?
- What is the physical interpretation of the constants found during integration?
Tip: Always apply initial conditions immediately after integration to simplify the process of finding constants.
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Math Problem Analysis
Mathematical Concepts
Calculus
Vector Calculus
Differential Equations
Kinematics
Formulas
v(t) = ∫a(t) dt
r(t) = ∫v(t) dt
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate (Calculus)
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