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

a(t)= 3i + 2j, v(0)= k, r(0)=i

find r(t).

To find the velocity and position vectors of a particle given its acceleration, initial velocity, and initial position, we can use the basic equations of motion.

First, let's integrate the acceleration vector to find the velocity vector. The integral of acceleration with respect to time gives us the change in velocity, so we have:

∫a(t) dt = ∫(3i + 2j) dt

Integrating each component separately, we get:

∫3i dt + ∫2j dt = 3∫i dt + 2∫j dt

The integral of dt is just t, so we have:

3∫i dt + 2∫j dt = 3ti + 2tj + C

Where C is the constant of integration. Now, we need to use the initial velocity value v(0)= k to find the specific value of C. Since the initial velocity is k and C is a constant, we can rewrite the equation as:

3(0)i + 2(0)j + C = ki

This simplifies to:

C = ki

Substituting this back into our equation for the velocity vector, we have:

v(t) = 3ti + 2tj + ki

Next, we can integrate the velocity vector to find the position vector. The integral of velocity with respect to time gives us the change in position, so we have:

∫v(t) dt = ∫(3ti + 2tj + ki) dt

Integrating each component separately, we get:

∫3ti dt + ∫2tj dt + ∫ki dt

Using the properties of integrals, we have:

3∫ti dt + 2∫tj dt + k∫i dt

The integral of ti is (1/2)ti^2, and the integral of tj is (1/2)tj^2. Since the particle starts at t=0, we can evaluate these integrals as:

(1/2)t^2i + (1/2)t^2j + K

Where K is a constant of integration. Now, we need to use the initial position value r(0) = i to find the specific value of K. Since the initial position is i and K is a constant, we can rewrite the equation as:

(1/2)(0)^2i + (1/2)(0)^2j + K = i

This simplifies to:

K = i

Substituting this back into our equation for the position vector, we have:

r(t) = (1/2)t^2i + (1/2)t^2j + i

Therefore, the position vector r(t) is:

r(t) = (1/2)t^2i + (1/2)t^2j + i