Question

A particle moves along the x-axis with velocity given by v(t) = 8cos(0.9t²) / (t² + 4t + 8) for time 0 ≤ t ≤ 15.

a. Find the acceleration of the particle at t.
b. Find an orthogonal matrix P and a diagonal matrix D such that PDP^T =
| 2 -1 0 0 |
| -1 2 0 0 |
| 0 0 2 -1 |
| 0 0 -1 2 |

Answer 1

The acceleration of a particle is obtained by differentiating the velocity function, and the particle's position is found by integrating the velocity function with appropriate initial conditions.

Step-by-step explanation:

a) The acceleration of a particle can be found by taking the derivative of its velocity function with respect to time. Given the velocity function v(t) = A + Bt¯¹, where A = 2 m/s and B = 0.25 m, the acceleration at a particular time t is calculated by a(t) = -Bt¯².

b) To find the particle's position at a given time t, one would need to integrate the velocity function. If the initial position is known, it can be used as the constant of integration.

For the provided velocity function, and with initial conditions that x(t = 1s) = 0, the position function x(t) at times t = 2.0 s and t = 5.0 s can be calculated by integrating the velocity function from the initial time to the desired time.