Final answer:
To find the velocity, acceleration, and speed of a particle, we need to differentiate the position function with respect to time and take the magnitude of the velocity vector. The velocity function is 4t i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k, the acceleration function is 4 i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k - 4t i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k, and the speed is ||v(t)|| = √(4t² + e⁽⁻²ᵗ⁾ + 16t² e⁽⁻⁴ᵗ⁾).
Step-by-step explanation:
To find the velocity, acceleration, and speed of a particle with the given position function r(t) = 2t² i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k, we need to differentiate the position function with respect to time.
The velocity function v(t) is obtained by taking the derivative of r(t) with respect to t. The acceleration function a(t) is obtained by taking the derivative of v(t) with respect to t. Finally, the speed is obtained by taking the magnitude of the velocity vector.
Taking the derivatives, we get: v(t) = 4t i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k and a(t) = 4 i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k - 4t i e⁽⁻ˣʸ⁾ j e⁽⁻²ᵗ⁾ k.
The speed is given by the magnitude of the velocity vector: ||v(t)|| = √(4t² + e⁽⁻²ᵗ⁾ + 16t² e⁽⁻⁴ᵗ⁾).