Question

Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = eᵗ (cos(t) i + sin(t)j + 8tk)

Answer 1

The velocity of a particle is obtained by differentiating its position function, acceleration by differentiating velocity, and speed is the magnitude of the velocity vector.

Step-by-step explanation:

To find the velocity of a particle with a given position function r(t) = et (cos(t) i + sin(t)j + 8tk), we differentiate the position vector with respect to time. The derivative gives us the velocity function v(t), which is:

• The velocity, v(t), is the first derivative of the position function. We apply the product rule of differentiation combined with the chain rule since et also depends on t.
• The acceleration, a(t), is the derivative of the velocity function v(t), which we get by differentiating v(t) with respect to time again.
• The speed is the magnitude of the velocity vector v(t), calculated using the Pythagorean theorem in the context of vector magnitudes.

For example, if we were to differentiate r(t) to find v(t), the i component would be d/dt [etcos(t)], the j component would be d/dt [etsin(t)], and the k component would be d/dt [et8t]. Similarly, acceleration is obtained by differentiating the velocity function resulting from the previous step.