Question

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 5 i + 8 j, v(0) = k, r(0) = i

Answer 1

To find the velocity and position vectors of a particle with the given acceleration, initial velocity, and initial position, you can integrate the acceleration function twice. The velocity vector is obtained by integrating the acceleration vector with respect to time, and the position vector is obtained by integrating the velocity vector with respect to time.

Step-by-step explanation:

To find the velocity and position vectors of a particle with the given acceleration, initial velocity, and initial position, we can integrate the acceleration function twice. The velocity vector is obtained by integrating the acceleration vector with respect to time, and the position vector is obtained by integrating the velocity vector with respect to time.

Given: a(t) = 5i + 8j, v(0) = k, r(0) = i

Step 1: Integrate a(t) to find v(t): ∫a(t)dt = ∫(5i + 8j)dt = 5ti + 8tj + C_1, where C_1 is the integration constant.

Step 2: Integrate v(t) to find r(t): ∫v(t)dt = ∫(5ti + 8tj + C_1)dt = (5/2)t^2i + (4/2)t^2j + C_1t + C_2, where C_2 is the integration constant.

Therefore, the velocity vector is v(t) = (5t)i + (8t)j + C_1, and the position vector is r(t) = (5/2)t^2 i + (4/2)t^2j + C_1t + C_2.