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### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

# 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?

The final velocity, $${v_f} = 22.4\,m/s$$, is the velocity of the falcon and dove together, after the collision along the positive x-direction (that is in the direction motion of falcon and dove before the collision).

See the step by step solution

## Step1: Writing the given data from the question.

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, $${v_2} = 7.00 m/s$$ ( v2 is 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 = vf

## Step 2: Explaining the theory of Law of conservation of momentum.

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 p1 and p2 be 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 vf or 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.

## Step 3: Finding the final velocity of the system 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 vf is the final velocity of the dove and falcon together.

Dividing both sides of the equation 3 by mt, gives

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

Where vf 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.