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Chapter 9: Problem 71

Two identical objects with the same initial speed collide and stick together. If the composite object moves with half the initial speed of either object, what was the angle between the initial velocities?

Short Answer

Expert verified
Using the principle of conservation of momentum and the dot product of the initial velocities, you will find that the angle between the initial velocities is 120 degrees.

Step by step solution

01

Write Down the Momentum Vectors

Initial momenta of the two objects can be represented as vectors: \(m\vec{v_1}\) and \(m\vec{v_2}\), and the final momentum is represented by \((2m)\vec{v_f}\) (since two objects of mass \(m\) combined have mass \(2m\)). Here \(\vec{v_1}\), \(\vec{v_2}\) are the velocities vectors of the two objects and \(\vec{v_f}\) is the velocity vector of the composite object.
02

Apply Conservation of Momentum

The conservation of momentum demands that the sum of the initial momenta be equal to the final momentum. That is, \(m\vec{v_1} + m\vec{v_2} = 2m\vec{v_f}\). We see that we can cancel \(m\) from both sides. The speeds are represented as \(|\vec{v_1}| = |\vec{v_2}| = |\vec{v_f}|\), since they are defined to be equal, we say \(v = |\vec{v_f}|\). From the given condition, we have \(v = 2|\vec{v_f}|\), where \|...\| expresses the magnitude of a vector.
03

Find the Angle between the Velocities

To find the angle between the velocities, we calculate the dot product of the initial velocities and use the geometrical formula: \(\vec{v_1}\cdot\vec{v_2} = |\vec{v_1}||\vec{v_2}|cos(\Theta)\), where \(\Theta\) is the angle between the velocities. After reorganizing, we find: \(cos(\Theta) = \frac{\vec{v_1}\cdot\vec{v_2}}{|\vec{v_1}||\vec{v_2}|}\). Lastly, the angle \(\Theta\) can be found by taking the arccosine (inverse of cosine) on both sides.

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