Final answer:
To find the minimum speed of a particle, we need to find the time at which the magnitude of the velocity vector is the smallest. This can be done by taking the derivative of the magnitude of the velocity with respect to time and setting it equal to zero. In this case, the time at which the speed is minimum is t = 0.384s.
Step-by-step explanation:
To find the minimum speed of a particle, we need to find the time at which the magnitude of the velocity vector is the smallest. The magnitude of the velocity vector can be found using the formula |v(t)| = sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2), where dx/dt, dy/dt, and dz/dt are the derivatives of the position function with respect to time. In this case, r(t) = 5t^2î + 5tĵ + (t^2 - 4t)k, so dx/dt = 10t, dy/dt = 5, and dz/dt = 2t - 4. Plugging these values into the formula, we have:
|v(t)| = sqrt((10t)^2 + 5^2 + (2t - 4)^2) = sqrt(104t^2 - 80t + 41)
To find the minimum of this expression, we can take the derivative of |v(t)| with respect to t and set it equal to zero, then solve for t. Taking the derivative, we get:
d(|v(t)|)/dt = (1/2)*(104t^2 - 80t + 41)^(-1/2)*(208t - 80) = 0
Simplifying this equation and solving for t, we find t = 0.384s.