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Magazine Chapter 9: Problem 73
After a completely inelastic collision, two objects of the same mass and same initial speed move away together at half their initial speed. Find the angle between the initial velocities of the objects.
Short Answer
Expert verified
The angle between their initial velocities is \( 120^{\circ} \).
Step by step solution
01
Understand the Problem
We are given two objects with the same mass, same initial speed, and after a completely inelastic collision, they move with half their initial speed. We need to find the angle between their initial velocities.
02
Define the Known Quantities
Let the mass of each object be \( m \) and their initial speed be \( v \). After the collision, the objects move together with a speed of \( \frac{v}{2} \).
03
Write the Momentum Conservation Equation
Before the collision, the momentum is \( m\vec{v}_1 + m\vec{v}_2 \). After the collision, the combined mass \( 2m \) moves with speed \( \frac{v}{2} \). Thus:
04
Set Up the Equation for Momentum Conservation
Using the conservation of momentum principle, we have:\[ m\vec{v}_1 + m\vec{v}_2 = 2m \cdot \frac{v}{2} \] Simplifying, we get \[ \vec{v}_1 + \vec{v}_2 = v \]
05
Use the Given Condition on Final Speed
Note that the resultant speed vector \( \frac{v}{2} \) implies that the vectors of object's velocities must add to a vector with half of that magnitude. Therefore the magnitude of the vector \( \vec{v}_1 + \vec{v}_2 \) must satisfy:\[ |\vec{v}_1| = |\vec{v}_2| = v \] Thus we have: \[ \sqrt{v^2 + v^2 + 2v^2\cos \theta} = v \] where \( \theta \) is the angle between the vectors.
06
Solve for the Angle \( \theta \)
The equation simplifies to: \[ \sqrt{2v^2(1 + \cos \theta)} = v \]Squaring both sides, we have: \[ 2v^2(1 + \cos \theta) = v^2 \]Simplifying gives: \[ 2(1 + \cos \theta) = 1 \]\[ 1 + \cos \theta = \frac{1}{2} \] \[ \cos \theta = -\frac{1}{2} \] \( \theta = 120^{\circ} \) since the cosine of \( 120^{\circ} \) is \(-\frac{1}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Momentum Conservation
In physics, the idea of momentum conservation is a fundamental principle that helps us understand collisions, like the inelastic collision in this exercise.
Simply put, momentum is the measure of the motion of an object and is the product of its mass and velocity. The law of conservation of momentum states that in a closed system, the total momentum before the collision equals the total momentum after the collision.
Here's how it applies:
Simply put, momentum is the measure of the motion of an object and is the product of its mass and velocity. The law of conservation of momentum states that in a closed system, the total momentum before the collision equals the total momentum after the collision.
Here's how it applies:
- Before the collision, the total momentum is given by the sum of the momenta of the individual objects. If object 1 has momentum \( m\vec{v}_1 \) and object 2 has momentum \( m\vec{v}_2 \), then the total initial momentum is \( m\vec{v}_1 + m\vec{v}_2 \).
- After the inelastic collision, these two objects stick together, forming a "combined mass" which carries a new velocity \( \frac{v}{2} \).
- The equation \( m\vec{v}_1 + m\vec{v}_2 = 2m \cdot \frac{v}{2} \) helps us balance the momentum before and after the collision.
Vector Addition
Vectors, such as velocities, have both magnitude and direction, making them more complex than simple scalar quantities. Vector addition involves combining these directional components to find a resultant vector.
In this problem, the initial velocities of two objects are vectors, which are added to solve their resultant speed after collision, with some critical conditions to consider:
In this problem, the initial velocities of two objects are vectors, which are added to solve their resultant speed after collision, with some critical conditions to consider:
- The vectors in question \( \vec{v}_1 \) and \( \vec{v}_2 \) have equal magnitudes since both objects have the same initial speed \( v \).
- To find the resultant velocity after collision, we apply vector addition: the sum \( \vec{v}_1 + \vec{v}_2 = v \) represents the condition that allows us to infer the angle between them.
- The process involves considering both the magnitudes and directions of the vector components, making angle calculations crucial in solving the vector addition problem.
Angle Between Velocities
Understanding the angle between velocities is central to solving problems involving vector addition and momentum.
In this context, the angle between the initial velocities \( \vec{v}_1 \) and \( \vec{v}_2 \) of the two colliding objects dictates the direction of the resulting vector after the collision.
In this context, the angle between the initial velocities \( \vec{v}_1 \) and \( \vec{v}_2 \) of the two colliding objects dictates the direction of the resulting vector after the collision.
- When vectors are added, if the "angle between them" is 120 degrees, the magnitude of the resultant vector represents half of the individual vectors' magnitudes, which matches the already known post-collision speed \( \frac{v}{2} \).
- This is because the cosine function, which deals with the angle in trigonometry relations, operates such that \( \cos(120^{\circ}) = -\frac{1}{2} \), explaining the reduction factor seen in vector magnitude.
- Solving for angle \( \theta \) involves using the cosine rule: solving \( \cos \theta = -\frac{1}{2} \) leads us to conclude \( \theta = 120^{\circ} \)}, revealing the geometrical nature of how vector angles affect the overall vector dynamics during collision.
