**Problem statement:**

A point particle is initially at rest at the position **r _{0}** =

**i**– 2

**j**(m) and his acceletation is given by

**a**= 3

**i**(m/s

^{2}). Find the particle’s position velocity and acceleration as functions of time.

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**Solution:**

First of all let’s draw the reference frame that we will use to solve the problem. We will also draw the initial conditions.

As you can see in the figure above, the positive direction of the axes can be represented in two different ways. You can use the unit vectors ** i ** and ** j ** that define the positive directions of the axes (in red in the figure) or indicate next to each coordinate the plus sign that determines the positive sense of the corresponding axis.It is not necessary to use both.

The unit vectors also indicate the unit on each axis. In the previous figure you have represented to scale the initial position vector (in green) and the acceleration of the particle.

The acceleration of the particle is a given in this problem. As you can see it is constant (the particle has the same acceleration at any moment of time):

To determine the velocity vector as a function of time, we use the definition of acceleration:

And then we integrate both members of the previous equation:

The integrals that appear in the above equation are definite integrals. In order to determine their upper and lower limits you should first look at what the integration variable is. In the second member of the equation it’s the time, so the integral will be evaluated between the initial time (which we will assume equal to zero) and a generic time t. In this way we will obtain the ** velocity vector as a function of time **.

The upper and lower limits of the integral in the first member must be two velocities since this is the integration variable. The lower limit is the velocity vector at the initial time (which is zero) and the upper limit is the velocity vector in a generic time t. Since the particle is initially at rest, we will replace this initial velocity with the null vector.

Integrating and substituting the limits of integration:

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To calculate the position vector as a function of time, we use its definition and integrate. The upper and lower limits of the integral are determined following the same procedure we used with the velocity vector:

Integrating and substituting the limits of integration:

And simplifying:

Do not forget to include the units in the results.

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