Question

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.

A.a(t) = (6t + et) i + 12t2 j
B.v(0) = 3i
C. r(0) = 4 i − 2 j

Answer 1

To determine the velocity and position vectors for a particle, one must integrate the given acceleration vector, and then integrate the resulting velocity while considering the initial conditions provided for velocity and position respectively.

Step-by-step explanation:

The question asks us to find the velocity and position vectors of a particle given its acceleration vector a(t) = (6t + et) i + 12t2 j, its initial velocity v(0) = 3i, and its initial position r(0) = 4i - 2j. To find the velocity vector v(t), we integrate the acceleration function with respect to time, taking into account the initial velocity:

1. Integrate each component of a(t) to find v(t):
   v(t) = ∫ a(t) dt = ∫ (6t + et) dt i + ∫ 12t2 dt j
2. Add the initial velocity to the result of the integration to find the constants of integration.
3. Integrate v(t) to find r(t), the position vector, again accounting for the given initial position.

The position vector r(t) is found by integrating the velocity vector v(t), using the initial position to find the constants of integration.