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A ball of mass \(m=1 \mathrm{~kg}\) is hung vertically by a thread of length \(l=1.50 \mathrm{~m}\). Upper end of the thread is attached to the ceiling of a trolley of mass \(M=4 \mathrm{~kg}\). Initially, the trolley is stationary and it is free to move along horizontal rails without friction. A shell of mass \(m=1 \mathrm{~kg}\), moving horizontally with velocity \(v_{0}=6 \mathrm{~m} / \mathrm{s}\) collides with the ball and gets stuck with it. As a result, the thread starts to deflect towards right. The velocity of the combined body just after collision is (1) \(2 \mathrm{~m} / \mathrm{s}\) (2) \(3 \mathrm{~m} / \mathrm{s}\) (3) \(\mid \mathrm{m} / \mathrm{s}\) (4) \(4 \mathrm{~m} / \mathrm{s}\)

Step by step solution

01

Understand the System Before Collision

Before the collision, the ball is at rest and the shell is moving horizontally with a velocity of \(v_0 = 6 \text{ m/s}\). The ball and shell have the same mass \(m = 1 \text{ kg}\). The trolley is stationary.

02

Apply Conservation of Linear Momentum

Momentum is conserved along the horizontal direction. Before collision, only the shell has momentum, while the ball and the trolley do not because they are at rest. The total momentum before the collision is \(m \cdot v_0 = 1 \cdot 6 = 6 \text{ kg m/s}\). After collision, the ball and shell stick together to form a new mass \(2m \text{ kg}\) and start moving with the velocity \(v\). There is no external force, so the momentum after collision is \(M \cdot V_t + 2m \cdot v\).

03

Solve for the Velocity after Collision

Since there are no external horizontal forces, the momentum is conserved:\[m \cdot v_0 = M \cdot V_t + 2m \cdot v\]Assuming the trolley does not affect initial momentum as it begins at rest, solving gives:\[1 \cdot 6 = 0 + 2 \cdot v \implies v = 3 \text{ m/s}\] This velocity is independent of the trolley since it is free to move without friction.

04

Conclude the Velocity of the Combined Body

The velocity of the combined body (ball and shell) immediately after the collision, from the conservation of linear momentum, is found to be \(v = 3 \text{ m/s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elastic Collision

An elastic collision is a type of collision where the total kinetic energy is conserved before and after the collision. In such collisions, both kinetic energy and momentum are conserved quantities. However, in the given exercise, we are dealing with a special case known as an inelastic collision, as described below. In a truly elastic collision, objects do not stick together, and they bounce off each other without any loss of energy in the form of heat or sound.
To identify an elastic collision, look for these characteristics:

• Conservation of kinetic energy: The total kinetic energy before collision equals the total kinetic energy after collision.
• Objects rebound off each other, with neither being deformed permanently.

This is quite different from inelastic collisions, where objects typically meld into one another, as you'll see in the next section.

Inelastic Collision

Inelastic collisions are more common in real-world scenarios. During an inelastic collision, the total kinetic energy is not conserved. Some kinetic energy is transformed into other forms of energy, like heat or sound. This is the type of collision we observe in the described exercise, where the shell and ball stick together after the collision. This feature of sticking together is characteristic of what is known as a "completely inelastic collision."
In the problem:

• Both the shell and ball have the same mass, making calculations straightforward.
• After the collision, the combined mass moves with a common velocity.
• Unlike an elastic collision, kinetic energy is lost due to the objects sticking together.

These attributes highlight the main difference between elastic and inelastic collisions.

Momentum Conservation

The law of momentum conservation is a fundamental principle in physics, stating that momentum is conserved in an isolated system, meaning the total momentum before a collision equals the total momentum after. In the exercise, this principle simplifies solving for the velocity of the combined shell and ball after the collision.
Key points about momentum conservation:

• Momentum is a vector quantity, having both magnitude and direction.
• It is expressed mathematically as \(p = mv\), where \(p\) is momentum, \(m\) is mass, and \(v\) is velocity.
• For the collision in this exercise, the momentum before is carried entirely by the shell.
• After collision, the momentum is distributed between the sticking ball and shell combination.

Applying this principle, we calculate the post-collision velocity as \(3 \text{ m/s}\). This constant foundational principle gives us a reliable method to predict the outcomes of collisions.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, calculated as \(KE = \frac{1}{2}mv^2\). Before the collision, only the moving shell holds kinetic energy. The ball, initially at rest, has none. However, after the collision, because it is inelastic, not all initial kinetic energy is retained in the system.
The distinctions of kinetic energy in inelastic collisions include:

• The initial kinetic energy is greater than the post-collision kinetic energy. Some energy is dissipated.
• This loss can manifest as heat, deformation, or sound.
• Conservation of kinetic energy only applies to elastic collisions.

While kinetic energy is not conserved in inelastic collisions, the principle provides critical insight into energy transformations occurring during and after the collision.