Question

The position vector r describes the path of an object moving in space. Find the velocity v(t), speed s(t), and acceleration a(t) of the object. Given r(t) = ti + t²j + t² - 2k?

Answer 1

To find the velocity, speed, and acceleration, differentiate the position vector to get the velocity v(t), calculate the magnitude of this velocity for speed s(t), and differentiate the velocity vector again to get acceleration a(t). In this case, the acceleration is constant since it does not depend on time t.

Step-by-step explanation:

Finding Velocity, Speed, and Acceleration

The position vector r(t) = ti + t²j + (t² - 2)k describes the path of an object in three-dimensional space with respect to time t. To find the velocity vector v(t), we differentiate each component of r(t) with respect to t.

V(t) = dr/dt = i + 2tj + 2tk

The speed s(t) is the magnitude of the velocity vector, which is given by

S(t) = |v(t)| = √((1)^2 + (2t)^2 + (2t)^2)

To find the acceleration vector a(t), we differentiate the velocity vector with respect to t.

A(t) = dv/dt = 2j + 2k

The acceleration vector a(t) shows that the acceleration is constant in time, as its components do not depend on t.