Impulse and change of direction

Question

A particle $P$, of mass $0.5$ kg, is moving with velocity $(4i+4j)$ m/s when it receives an impulse $I$ of magnitude $2.5$ Ns.

As a result of the impulse, the direction of motion of $P$ is deflected through an angle of $45^\circ$.

Given that $I=(\lambda i + \mu j)$ Ns, find all the possible pairs of $\lambda$ and $\mu$.

I've tried to draw a diagram to visualise the problem however each time i do this, I'm getting $\mu$ to be $0$.

From what I understand this is not possible, could someone point me in the right direction?

I've been stuck on this problem for an hour now.

Answer 1

Based on your comment, what I can gather is that maybe the question was misinterpreted. Here, you are given that the final velocity is $45 ^o $ with respect to the initial velocity. This gives us two cases for the final velocity - either along x axis or y axis. I'll demonstrate one case here

Let final velocity be $v_f = a$i $$\textbf I = m\Delta\textbf v$$ $$\textbf I = 0.5(a \textbf i - (4\textbf i + 4\textbf j))$$ $$\textbf I = 0.5((a-4)\textbf i - 4\textbf j)$$

Now, using the fact that $|\textbf I| = 2.5 $ Ns, you can obtain $a$, resubstitute to get one pair of $(\lambda, \mu)$.

Repeat with $v_f = a \textbf j$

Answer 2

Initial velocity is $u=4i+4j$. You have correctly identified this as travelling at an angle of $45^\circ$ to the horizontal.

Final velocity is either $v=ai$ (horizontal) or $v=bj$ (vertical). $a>0$ and $b>0$.

Consider the case where $v=ai$

Change in momentum is $m(v-u)$

$\lambda i + \mu j=0.5(ai-4i-4j)$

$\lambda i + \mu j=(0.5a-2)i-2j$

Vertical components: $\mu=-2$

Horizontal components: $\lambda=0.5a-2$

But $\lambda^2+\mu^2=2.5^2$

$(0.5a-2)^2=6.25-4$

$0.5a-2=\sqrt {2.25}$

$0.5a-2=1.5$ or $0.5a-2=-1.5$

$0.5a=3.5$ or $0.5a=0.5$

$a=7$ or $a=1$

Then $\lambda=3.5-2=1.5$ or $\lambda=0.5-2=-1.5$

Do similarly with the other case, where $v=bj$.

Change in momentum is $m(v-u)$

$\lambda i + \mu j=0.5(bj-4i-4j)$

$\lambda i + \mu j=-2i+(0.5b-2)j$

Vertical components: $\mu=0.5b-2$

Horizontal components: $\lambda=-2$

But $\lambda^2+\mu^2=2.5^2$

$(0.5b-2)^2=6.25-4$

$0.5b-2=\sqrt {2.25}$

$0.5b-2=1.5$ or $0.5b-2=-1.5$

$0.5b=3.5$ or $0.5b=0.5$

$b=7$ or $b=1$

Then $\mu=3.5-2=1.5$ or $\mu=0.5-2=-1.5$