A\(1.80 kg\)falcon catches a\(0.650 kg\)dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially\( 28.0 m/s\)and the dove’s velocity is\(7.00 m/s\)in the same direction?

Given:

The initial velocity of the falcon,\({v_1} = 28.0 m/s\)

Mass of the falcon,\({m_1} = 1.80 kg\)

Mass of the dove,\({m_2} = 0.650 kg\)

The initial velocity of the dove,\({v_2} = 7.00 m/s\)( v₂is positive because the dove is having the same direction as the falcon after catching the dove)

Total mass after the collision,\({m_t} = {m_1} + {m_2}\)

{Since the masses (falcon and dove) stick together after the collision}

The final velocity of the falcon = The final velocity of the dove = v_f

The Law of conservation of momentum states that the total momentum of a system always remains constant before and after collisions or we can say that the initial momentum before the collision of a system is equal to the final momentum of the system after the collision.

Let p₁and p₂be the initial momentum of two objects before the collision,\({p'_1}\)and\({p'_2}\)be the final momentum after the collision, then according to the law of conservation of momentum,

\({p_1} + {p_2} = {p'_1} + {p'_2}\).................(1)

In this problem, after the collision, the two masses (mass of falcon and dove) move with a final velocity v_for the masses get to stick together. These types of collisions are known as inelastic collisions.

Therefore, equation 1 becomes\({p_1} + {p_2} = {p'_t}\).........(2)

Where \({p'_t}\) is the final momentum of the falcon and deer together.

Substituting the value of\(p = mv\)in equation 2, we get

\({m_1}{v_1} + {m_2}{v_2} = {m_t}{v_f}\).........(3)
where v_f is the final velocity of the dove and falcon together.

Dividing both sides of the equation 3 by m_t, gives

\(\dfrac{{{m_1}{v_1} + {m_2}{v_2}}}{{{m_t}}} = {v_f}\)

Where v_f is the final velocity of the system together.

\({v_f} = \dfrac{{{m_1}{v_1} + {m_2}{v_2}}}{{{m_t}}}............(4)

Substitute the values in equation 4, we get

\({v_f} = \dfrac{{(1.80kg \times 28m/s) + (0.650kg \times 7.00m/s)}}{{1.80kg + 0.650kg}}\)

\(\begin{aligned} &= \dfrac{{(1.80kg \times 28m/s) + (0.650kg \times 7.00m/s)}}{{1.80kg + 0.650kg}}\\ &= \dfrac{{(50.4kg \cdot m/s) + (4.55kg \cdot m/s)}}{{2.45kg}}\\ &= \dfrac{{54.95kg \cdot m/s}}{{2.45kg}}\\ &= 22.4m/s\end{aligned}\)

Thus the final velocity, \({v_f} = 22.4\,m/s\), is the velocity of the falcon and dove together, after the collision.

Hence, the final velocity is positive indicates that the system will move together along the positive x-direction after the collision or after catching the dove by the falcon.