## Statement

If a body moves according to the following equation:

$$\overrightarrow{r\left(t\right)}=(4\xb7t+{t}^{2})\xb7\overrightarrow{i}+4\xb7t\xb7\overrightarrow{j}m$$

Calculate its instantaneous velocity at time t=1 s.

## Solution

**Data**

$$\overrightarrow{r\left(t\right)}=(4\xb7t+{t}^{2})\xb7\overrightarrow{i}+4\xb7t\xb7\overrightarrow{j}m$$

**Resolution**

To solve this problem, we will use the following equation, that establishes that the instantaneous velocity is the derivative of the position vector with respect to time.

Differentiating or calculating the limit, we get that:

$$\overrightarrow{v\left(t\right)}=(4+2\xb7t)\xb7\overrightarrow{i}+4\xb7\overrightarrow{j}m$$

Once we know the instantaneous velocity vector, we substitute the value of t=1 s and we get the instantaneous velocity for said time:

$$\overrightarrow{v\left(1\right)}=(4+2\xb71)\xb7\overrightarrow{i}+4\xb7\overrightarrow{j}m\Rightarrow \phantom{\rule{0ex}{0ex}}\overline{)\overrightarrow{v\left(1\right)}=6\xb7\overrightarrow{i}+4\xb7\overrightarrow{j}m}$$