Problem 4 - Applied Mathematics

Question

Two masses of 20 and 10 units. moving in the same direction at speeds of 16 and 12 units respectively collide and stick together. Find the velocity of the combined mass immediately afterwards.

Step-by-Step Solution

Answer

The velocity of the combined mass immediately after the collision is 14.6 units.

Step 1 - Identify Given Values

Identify the masses and velocities of the objects involved. Let the masses be \( m_1 = 20 \) units and \( m_2 = 10 \) units, and their velocities be \( v_1 = 16 \) units and \( v_2 = 12 \) units respectively.

Step 2 - Write Down the Conservation of Momentum

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. The equation is given by: \[ m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_f \]

Step 3 - Substitute the Values into the Equation

Substitute the given values into the momentum equation. \[ 20 \times 16 + 10 \times 12 = (20 + 10)v_f \] \[ 320 + 120 = 30v_f \]

Step 4 - Solve for the Final Velocity

Combine the terms on the left side and then solve for \( v_f \). \[ 440 = 30v_f \] Divide both sides by 30: \[ v_f = \frac{440}{30} \] \[ v_f = 14.\bar{6} \]

Key Concepts

momentum

Momentum is a fundamental concept in physics. It quantifies the amount of motion an object has and is given by the product of its mass and velocity. The formula for momentum is: \[ p = mv \] where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. In our exercise, we dealt with two objects, having masses of 20 and 10 units and velocities of 16 and 12 units respectively. By understanding momentum, you can predict how objects will move after interacting.

collision

In physics, a collision happens when two or more bodies exert forces on each other for a relatively short time. There are different types of collisions, such as elastic (where objects bounce off each other) and inelastic (where objects stick together). In this exercise, we are dealing with an inelastic collision because the two masses stick together after colliding. Understanding the type of collision helps in applying the right principles and equations to calculate the resulting motion.

velocity calculation

Calculating velocity is essential in solving many physics problems. In our exercise, we needed to find the velocity of the combined mass after the collision. Using the conservation of momentum formula: \[ m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_f \] we can substitute the given values and solve for \( v_f \). After substituting, we get: \[ 20 \times 16 + 10 \times 12 = 30v_f \] \[ 440 = 30v_f \] By dividing both sides by 30, we find that the final velocity \( v_f \) is approximately 14.\bar{6} units. This shows how the velocities and masses of objects before a collision determine their combined velocity after the collision.

physics

Physics involves studying the natural laws that govern the behavior of matter and energy. Fundamental principles like the conservation of momentum allow us to understand and predict the outcomes of different scenarios. In this exercise, the law of conservation of momentum helped us determine the final velocity of two colliding masses. Knowing these principles not only helps with academic problems but also in practical applications in real life like car crashes, sports, and even space explorations.