93. Two particles each of mass 20 g are involved in a one-dimensional elastic collision. The first particle moves at 10 m s⁻¹ while the second is initially at rest. Find the speed of the first particle after collision.
One-Dimensional Elastic Collision Between Two Equal Masses
Collisions are one of the most important topics in Classical Mechanics because they explain how objects interact when they strike one another. From collisions between billiard balls and vehicles to molecular collisions in gases and particle interactions in nuclear physics, the principles of momentum and energy conservation govern every collision.
Correct Answer
0 m/s
Understanding Elastic Collision
An elastic collision is a collision in which both the total linear momentum and the total kinetic energy of the system remain conserved. Unlike an inelastic collision, no kinetic energy is lost as heat, sound, or deformation.
Therefore, two conservation laws are simultaneously applicable:
- Conservation of Linear Momentum
- Conservation of Kinetic Energy
Because both laws are satisfied, elastic collision problems can be solved exactly using algebra.
Characteristics of a One-Dimensional Elastic Collision
In a one-dimensional collision, both particles move along the same straight line before and after the collision. Since motion occurs along only one axis, vector equations reduce to simple algebraic equations involving positive and negative signs.
For an elastic collision:
- Total momentum before collision = Total momentum after collision.
- Total kinetic energy before collision = Total kinetic energy after collision.
- The relative speed of separation equals the relative speed of approach.
These three conditions completely describe the motion after the collision.
Given Data
Mass of first particle
m₁ = 20 g = 0.02 kg
Mass of second particle
m₂ = 20 g = 0.02 kg
Initial velocity of first particle
u₁ = 10 m/s
Initial velocity of second particle
u₂ = 0 m/s
Required:
Final velocity of the first particle (v₁)
Step 1: Apply Conservation of Momentum
The total linear momentum before collision is
m₁u₁ + m₂u₂
After collision, the total momentum is
m₁v₁ + m₂v₂
Therefore,
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Since both masses are equal,
u₁ + u₂ = v₁ + v₂
Substituting the given values,
10 + 0 = v₁ + v₂
v₁ + v₂ = 10 ………… (1)
Step 2: Apply Conservation of Kinetic Energy
Since the collision is elastic,
½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²
Again, because the masses are equal,
u₁² + u₂² = v₁² + v₂²
Substituting the values,
10² + 0² = v₁² + v₂²
100 = v₁² + v₂² ………… (2)
Step 3: Solve the Two Equations
From Equation (1),
v₂ = 10 − v₁
Substitute this into Equation (2).
100 = v₁² + (10 − v₁)²
Expanding,
100 = v₁² + 100 − 20v₁ + v₁²
0 = 2v₁² − 20v₁
2v₁(v₁ − 10) = 0
This gives two mathematical solutions:
v₁ = 10 m/s
or
v₁ = 0 m/s
The first solution represents the situation in which no collision occurs, so it is not physically relevant.
Hence, the actual physical solution is
v₁ = 0 m/s
Shortcut Formula for Equal Masses
For a one-dimensional perfectly elastic collision between two particles of equal mass, the particles simply exchange their velocities after the collision.
Initially,
Particle 1 → 10 m/s
Particle 2 → 0 m/s
After collision,
Particle 1 → 0 m/s
Particle 2 → 10 m/s
This shortcut is extremely useful in competitive examinations and saves valuable time.
Physical Interpretation
Imagine two identical smooth billiard balls. One ball is moving while the other is stationary. During a perfectly elastic head-on collision, the moving ball transfers all of its momentum and kinetic energy to the stationary ball. As a result, the first ball comes to rest, and the second ball moves away with exactly the same speed that the first ball initially had.
This behavior is commonly observed in a Newton’s cradle, where identical steel balls transfer momentum and kinetic energy almost perfectly through elastic collisions.
Real-Life Applications
The principles of elastic collisions are applied in billiards, Newton’s cradle demonstrations, molecular collisions in gases, particle accelerators, nuclear scattering experiments, and many areas of engineering and astrophysics. Understanding these collisions helps scientists analyze interactions ranging from microscopic atoms to massive celestial bodies.
Exam-Oriented Key Concepts
Students should remember that in a one-dimensional elastic collision between two identical masses, the velocities are exchanged after the collision. This result follows directly from the simultaneous conservation of momentum and kinetic energy. Whenever one identical particle is initially at rest, the moving particle stops after the collision, while the stationary particle moves away with the original speed of the first particle. This is one of the most frequently tested concepts in mechanics.
Final Answer
After the one-dimensional elastic collision, the first particle comes to rest.
Speed of the first particle after collision = 0 m/s
Final Answer: 0 m/s
