Impulse

Data

$\overrightarrow{F}=-6 ·\overrightarrow{i} +8·\overrightarrow{j} N$
$\overrightarrow{v_o}=15 ·\overrightarrow{i} -10·\overrightarrow{j} N$
Δt = 5 s
m = 1 kg
Linear momentum before applying the force p₀ = ?
Linear momentum after applying the force p = ?

Resolution

Question a)

We know impulse of a force is calculated by multiplying that force by the time  applied, so:

$$\begin{gathered} \overrightarrow{J}=\overrightarrow{F}·∆t \Rightarrow \\ \overrightarrow{J}=\left(-6 ·\overrightarrow{i} +8·\overrightarrow{j}\right) N · 5 s \Rightarrow \\ \boxed{\overrightarrow{J}=-30 · \overrightarrow{i}+40 · \overrightarrow{j} N·s} \end{gathered}$$

Question b)

By using the linear momentum definition we can calculate the moment before applying the force:

Since the impulse is equal to the growth of the linear momentum when applying the force, and the growth is the substraction of momentum the body has before and after applying the force:

$$\begin{gathered} \overrightarrow{J}=∆\overrightarrow{p} = \overrightarrow{p}-{\overrightarrow{p}}_0\Rightarrow \\ \overrightarrow{p}=\overrightarrow{J}+{\overrightarrow{p}}_0 \Rightarrow \\ \overrightarrow{p}=\left(-30·\overrightarrow{i}+40·\overrightarrow{j} N·s \right)+\left(15·\overrightarrow{i}-10·\overrightarrow{j}N·s\right)\Rightarrow \\ \boxed{\overrightarrow{p}=-15·\overrightarrow{i} + 30 ·\overrightarrow{j} N·s} \end{gathered}$$