Final answer:
The velocity of the particle is the first derivative of the position function, the acceleration is the second derivative, and the speed is the magnitude of the velocity vector.
Step-by-step explanation:
To find the velocity, acceleration, and speed of a particle with the given position function r(t) = 2t² i + e⁴²⁴ j + e⁴⁹⁴⁹ k, you need to take the derivatives of the position function with respect to time.
First, to find velocity (v(t)), differentiate each component of the position vector r(t):
- v(t) = dr(t)/dt = 4t i + 4e⁴²⁴ j - 4e⁴⁹⁴⁹ k
Next, to find acceleration (a(t)), differentiate the velocity function:
- a(t) = dv(t)/dt = 4 i + 8e⁴²⁴ j - 8e⁴⁹⁴⁹ k
Finally, the speed at any point in time is the magnitude of the velocity vector. At time t:
- Speed(t) = |v(t)| = √(4t)² + (4e⁴²⁴)² + (-4e⁴⁹⁴⁹)²