Find the speed of a particle with the given position function.
r(t)=√2 i +e^t j +e^-t k
v(t) = dr/dt = e^t j - e^-t k
speed = |v| = √(e^2t + e^-2t)
To find the speed of a particle with the given position function r(t), we need to differentiate the function with respect to t to obtain the velocity vector. Then, we calculate the magnitude of the velocity vector to determine the speed.
The given position function is r(t) = √2 i + e^t j + e^-t k.
To find the velocity vector, we differentiate each component of r(t) with respect to t:
v(t) = dr(t)/dt = d(√2 i)/dt + d(e^t j)/dt + d(e^-t k)/dt
Differentiating each component:
v(t) = 0 i + e^t j - e^-t k
Now, we can calculate the magnitude of the velocity vector to find the speed:
Speed = |v(t)| = √(0^2 + (e^t)^2 + (-e^-t)^2)
= √(e^(2t) + e^(-2t))
Therefore, the speed of the particle with the given position function r(t) is √(e^(2t) + e^(-2t)).