A particle of mass m1, traveling at speed v0, collides head-on with a particle of mass m2 at rest. The collision is perfectly inelastic, i.e., the particles fuse together at the collision. Calculate the fraction of the initial kinetic energy dissipated in the collision. Assume the

collision is Newtonian, i.e., mass is conserved.

KE=0.5MV

To calculate the fraction of the initial kinetic energy dissipated in the collision, we need to first determine the final velocity of the particles after the collision. Since the collision is perfectly inelastic and the particles fuse together, they will move together with a common final velocity.

To solve this problem, we can use the principle of conservation of momentum and conservation of kinetic energy.

1. Conservation of Momentum:
According to the principle of conservation of momentum, the total momentum of the system before the collision is equal to the total momentum after the collision. Mathematically, we can write it as:

(m1 * v0) + (m2 * 0) = (m1 + m2) * vf

Where:
m1 = mass of the first particle
m2 = mass of the second particle
v0 = initial velocity of the first particle
vf = final velocity of the combined particles

Since the second particle is at rest initially, its momentum is zero.

2. Conservation of Kinetic Energy:
The initial kinetic energy of the system is given by:

KE_initial = 0.5 * m1 * v0^2

The final kinetic energy of the combined particles is given by:

KE_final = 0.5 * (m1 + m2) * vf^2

Since the collision is perfectly inelastic, some of the initial kinetic energy dissipates. Thus, the fraction of initial kinetic energy dissipated can be calculated as:

Fraction of energy dissipated = (KE_initial - KE_final) / KE_initial

Now, let's calculate the final velocity and the fraction of energy dissipated.

From the conservation of momentum equation:

(m1 * v0) = (m1 + m2) * vf

Simplifying this equation, we can find the value of vf:

vf = (m1 * v0) / (m1 + m2)

Now substituting this value of vf into the equation for final kinetic energy:

KE_final = 0.5 * (m1 + m2) * [(m1 * v0) / (m1 + m2)]^2

The fraction of energy dissipated is then:

Fraction of energy dissipated = (KE_initial - KE_final) / KE_initial

Substituting the values, you can find the numerical value of the fraction of energy dissipated.

Remember to their units for mass (e.g., kilograms) and velocity (e.g., meters per second) consistent.

I hope this explanation helps you solve the problem!

find the resultant velocity with the conservation of momentum

m1*vo=(m1+m2)V

solve for V

determine finalKE, and initial KE, and the fracation.

the difference between final and initial KE is lost ke, the fraction would be that amount lost divided by initial