Collision and conservation of mechanical energy – probems and solutions

1. Two objects have the same mass, m_{1} = m_{2 }= 0.5 kg dropped from the same height as shown in the figure below. The radius of the circle is 1/5 m. The collision between both objects is perfectly elastic. Determine the velocity of each object after the collision. Acceleration due to gravity is 10 m/s^{2}.

__Known :__

Mass of object (m) = m_{1 }= m_{2} = 0.5 kg

Initial height (h_{1}) = 1/5 m

Final height (h_{2}) = 0 *(**base of path**)*

Initial speed of object (v_{1}) = 0 *(**object initially at rest**)*

Final speed of object (v_{2}) = …. *(**speed of object at base of path** = **speed of object before collision**)*

Acceleration due to gravity (g) = 10 m/s

__Wanted :__ Speed of each object after collision

__Solution :__

**Speed of object before collision**

Speed of object before collision = speed of object when arrive at base of path = the final speed of object.

*Initial mechanical energy** = **final mechanical energy*

*The gravitational potential energy **+ **kinetic energy** = **the gravitational potential energy **+ **kinetic energy*

*m g h*_{1}* + 1/2 m v*_{1}^{2 }*= m g h*_{2}* + 1/2 m v*_{2}^{2 }

*m g h*_{1}* + 0** **= 0 + 1/2 m v*_{2}^{2 }

*m g h*_{1 }*= 1/2 m v*_{2}^{2 }

*g h*_{1} *= 1/2 v*_{2}^{2 }

*2 g h*_{1 }*= v*_{2}^{2 }

*2(10)(1/5) = v*_{2}^{2 }* *

*2(2) = v*_{2}^{2 }* *

*4 = v*_{2}^{2}

*v*_{2 }*= **√4*

*v*_{2} *= **2 m/s*

*Speed of each object before collision is **2 m//s.*

**The speed of object after collision**

If both objects have the same mass and move in opposite direction then when collide, both objects change its velocity. For example, if before collision object A moves at 2 m/s and object B moves at -4 m/s, after collision object A moves at 4 m/s and object B moves -2 m/s. Minus and plus sign indicates that both objects have the different direction.

2. Mass of object A is 2-kg and mass of object B is 3 kg dropped from a height as shown in the figure below. Both objects collide at point C. The collision is perfectly elastic. Acceleration due to gravity is 10 m/s^{2}. Determine the speed of object A and speed of object B after the collision.

__Known :__

Mass of object A (m_{1}) = 2 kg

Mass of object B (m_{2}) = 3 kg

Initial height (h_{1}) = 5 meters

Final height (h_{2}) = 0 *(**base of path**)*

Initial speed of the object (v_{1}) = 0 *(**initially object at rest**)*

Final speed of the object (v_{2}) = …. *(**speed at base of path** = **speed before collision**)*

Acceleration due to gravity (g) = 10 m/s

__Wanted :__ Speed of each object after collision

__Solution :__

**Speed of object before collision**

Speed of object before collision = speed of object at base of path

*The initial mechanical energy** = **the final mechanical energy*

*The gravitational potential energy** = **kinetic energy*

*m g h*_{1 }*= 1/2 m v*_{2}^{2 }

*g h*_{1 }*= 1/2 v*_{2}^{2 }

*2 g h*_{1} *= v*_{2}^{2 }

*2(10)(5) = v*_{2}^{2 }* *

*100 = v*_{2}^{2 }* *

*v*_{2 }*= **√100*

*v*_{2 }*= 10 **m/s*

*Speed of each object before collision **= 10 m/s. *

**Speed of object after collision**

Both objects have the different mass and moves in different direction so speed of each object after collision calculated using this equation.

The speed of each object just after the collision :

**What is the difference between an elastic and an inelastic collision?****Answer:**In an elastic collision, both kinetic energy and momentum are conserved. In an inelastic collision, momentum is conserved, but kinetic energy is not. In a perfectly inelastic collision, the objects stick together after the collision.

**How is momentum conserved in a collision, regardless of whether it’s elastic or inelastic?****Answer:**Momentum is conserved in a collision due to Newton’s third law, which states that for every action, there’s an equal and opposite reaction. The total momentum of the system before the collision is equal to the total momentum after the collision.

**Why is kinetic energy not always conserved in a collision?****Answer:**Kinetic energy is not conserved in inelastic collisions because some of the initial kinetic energy is converted into other forms of energy, such as sound, heat, or deformation of the objects.

**Can the total mechanical energy of a system change during a collision? If so, how?****Answer:**Yes, the total mechanical energy can change during an inelastic collision. Although the total mechanical energy (kinetic plus potential) is conserved in an isolated system, in an inelastic collision, some kinetic energy may be transformed into non-mechanical forms like heat or sound.

**How can one determine if a collision is elastic or inelastic just by observing the objects before and after the collision?****Answer:**If the objects bounce off each other and the total kinetic energy before the collision is equal to the total kinetic energy after the collision, it’s an elastic collision. If the objects stick together or the total kinetic energy decreases, it’s an inelastic collision.

**What is the role of the coefficient of restitution in analyzing collisions?****Answer:**The coefficient of restitution (usually denoted by $e$) is a measure of how “bouncy” a collision is. It’s defined as the relative speed of separation divided by the relative speed of approach. For a perfectly elastic collision, $e=1$, and for a perfectly inelastic collision, $e=0$.

**How does the conservation of angular momentum apply to collisions?****Answer:**In a collision where there’s no external torque acting on the system, angular momentum is conserved. This can apply to objects spinning and colliding or to situations like ice skaters pulling their arms in to spin faster.

**Why might real-world collisions often appear to be inelastic?****Answer:**Real-world collisions typically involve some loss of kinetic energy to sound, heat, or deformation. These energy transformations make most real-world collisions inelastic to some degree.

**In a one-dimensional elastic collision between two objects with the same mass, what happens to their velocities after the collision?****Answer:**In a one-dimensional elastic collision between two objects of equal mass, their velocities simply switch after the collision. Object 1 ends with the initial velocity of Object 2, and vice versa.

**Can two objects stick together in an elastic collision?**

**Answer:**No, if two objects stick together, the collision is perfectly inelastic. In an elastic collision, the objects must rebound from one another, and the total kinetic energy must be conserved.