Problem 62 - University Physics with Modern Physics

Question

Two balls of equal mass collide and stick together as shown in the figure. The initial velocity of ball \(\mathrm{B}\) is twice that of ball A. a) Calculate the angle above the horizontal of the motion of mass \(\mathrm{A}+\mathrm{B}\) after the collision. b) What is the ratio of the final velocity of the mass \(A+B\) to the initial velocity of ball \(A, v_{f} / v_{A} ?\) c) What is the ratio of the final energy of the system to the initial energy of the system, \(E_{\mathrm{f}} / E_{\mathrm{i}}\) ? Is the collision elastic or inelastic?

Step-by-Step Solution

Answer

Question: After a collision between two balls A and B of equal mass, where ball B has an initial velocity twice that of ball A, find a) the angle above the horizontal of the motion of the combined mass (A + B), b) the ratio of the final velocity of the mass A+B to the initial velocity of ball A, and c) the ratio of the final energy of the system to the initial energy, and determine if the collision is elastic or inelastic. Answer: a) The angle above the horizontal of the motion of the combined mass (A + B) after the collision is given by: \(\beta = \arctan(\frac{2p_A\sin\alpha}{p_A+2p_A\cos\alpha})\) b) The ratio of the final velocity of the mass A+B to the initial velocity of ball A is given by: \( \frac{v_{final}}{v_A} = \frac{\sqrt{(mv_{final_x})^2+(mv_{final_y})^2}}{mv_A} \) c) The ratio of the final energy of the system to the initial energy is: \( \frac{E_{f}}{E_{i}} = \frac{\frac{1}{2}(2m)v_{final}^2}{\frac{1}{2} m v_A^2 + \frac{1}{2} m (2v_A)^2} \) If the ratio \(E_{f}/E_{i}\) is equal to 1, the collision is elastic; otherwise, it is inelastic.

Step 1: Find the Linear Momentum of Each Ball Before the Collision

We will use the linear momentum formula: \(p = mv\), where \(p\) is linear momentum, \(m\) is mass, and \(v\) is velocity. Since both balls have equal mass \(m\): \(p_A = mv_A\) \(p_B = m(2v_A)\)

Step 2: Find the Total Initial Linear Momentum of the System

Now, we will find the total linear momentum for the system by vector addition of \(p_A\) and \(p_B\). Let's denote the x-axis as \(\mathrm{horizontal}\) and y-axis as \(\mathrm{vertical}\). The sum of linear momenta is: \(p_x = p_A+p_B\cos\alpha\) \(p_y = p_B\sin\alpha\) Since \(p_B = 2p_A\), we get: \(p_x = p_A+2p_A\cos\alpha\) \(p_y = 2p_A\sin\alpha\)

Step 3: Find the Linear Momentum of the Combined Mass After the Collision

By the conservation of linear momentum, the total initial momentum should be equal to the total final momentum after the collision. \(p_{final_x} = p_x\) \(p_{final_y} = p_y\) Since the balls stick together, their combined mass is \(2m\). \(mv_{final_x} = p_A+2p_A\cos\alpha\) \(mv_{final_y} = 2p_A\sin\alpha\)

Step 4: Find the Angle Above the Horizontal

Now, we can find the angle \(\beta\) above the horizontal of the motion of mass A+B after the collision using the arctangent function: \(\beta = \arctan(\frac{v_{final_y}}{v_{final_x}})\) As a result, \(\beta = \arctan(\frac{2p_A\sin\alpha}{p_A+2p_A\cos\alpha})\)

Step 5: Find the Ratio of Final Velocity to Initial Velocity

We can calculate the final velocity for the combined mass:\( v_{final} = \sqrt{v_{final_x}^2 + v_{final_y}^2}\) Using the results from step 3: \(v_{final} = \sqrt{(mv_{final_x})^2+(mv_{final_y})^2}/m\) Now we can find the ratio \(v_{final}/v_A\).

Step 6: Calculate the Ratio of Final Energy to Initial Energy

For this step, we need to calculate the initial and final kinetic energies. The initial kinetic energy is given by: \(E_{i} = \frac{1}{2} m v_A^2 + \frac{1}{2} m (2v_A)^2\) The final kinetic energy is given by: \(E_{f} = \frac{1}{2}(2m)v_{final}^2\) Now, we can find the ratio: \(E_{f}/E_{i}\).

Step 7: Determine Whether the Collision is Elastic or Inelastic

If the ratio of final energy to initial energy is equal to 1, the collision is elastic; otherwise, it is inelastic. Compare \(E_{f}/E_{i}\) to 1 to determine the type of collision.

Key Concepts

Conservation of Linear Momentum

Understanding the conservation of linear momentum is crucial when analyzing collisions. This principle states that, within a system where no external forces are acting, the total linear momentum before an event must equal the total linear momentum after the event. For our exercise involving two balls, we apply this law to calculate the momentum after they stick together post-collision.

Imagine you're observing a game of billiards, where a cue ball strikes another, they move apart after impact. In a frictionless world, the cue ball's initial momentum and the second ball's subsequent momentum are equal. Translating this to our scenario with balls A and B: we start by calculating their individual momenta, then, we add them vectorially as they collide and stick together. The resulting total momentum remains the same as the sum of their initial momenta. This rule ensures that, regardless of the masses or velocities, the momentum of the isolated system is conserved.

Elastic and Inelastic Collisions

Collisions are categorized into two main types: elastic and inelastic.

In an elastic collision, both momentum and kinetic energy are conserved. Here, the colliding entities rebound with no energy loss. Think of two perfectly elastic balls bouncing off each other - they would keep their speed post-collision, or more technically, kinetic energy would be the same before and after.

Conversely, an inelastic collision is where the entities do not conserve kinetic energy, though momentum is still conserved. They might stick together or deform, losing some energy to sound or heat. In the problem at hand, after balls A and B collide and stick together, they must have lost some kinetic energy because their movements post-collision are different from those pre-collision, hinting that it's an inelastic collision. To confirm this, you would compare the ratios of final to initial energies.

Kinetic Energy in Collisions

Kinetic energy, the energy possessed by a body due to its motion, is a key player in the theatre of collisions. Its amount in a closed system may or may not be conserved, depending on the type of collision.

When we calculate the kinetic energy of objects before and after a collision, we gain insight into the nature of the collision. In our original exercise, we observe that balls A and B hail with certain velocities, carrying kinetic energies that, upon collision and sticking together, transform into another form of kinetic energy for the combined mass. However, due to the nature of inelastic collisions, there is always some loss. The final kinetic energy is less than the sum of the initial kinetic energies of the two balls. To assess this change quantitatively, the ratio of final to initial kinetic energies is determined, which also allows us to assert the non-elastic nature of the collision.