A 2.0 kg ball moving with a speed of 3.0 m/s hits, elastically, an identical stationary ball as shown. If the first ball moves away with angle 30° to the original path, determine
the speed of the first ball after the collision.
the speed and direction of the second ball after the collision.
To determine the speed of the first ball after the collision, we can use the principle of conservation of linear momentum. This principle states that the total momentum before the collision is equal to the total momentum after the collision, provided there are no external forces acting on the system.
Let's denote the initial velocity of the first ball as v1i, and the velocity of the second ball (initially stationary) as v2i. After the collision, the first ball moves away with a speed v1f at an angle of 30° to its original path. The second ball moves with a speed v2f.
First, let's calculate the initial momentum of the system:
Momentum before collision = (mass of the first ball * initial velocity of the first ball) + (mass of the second ball * initial velocity of the second ball)
Initial momentum = (2.0 kg * 3.0 m/s) + (2.0 kg * 0 m/s)
Initial momentum = 6.0 kg⋅m/s
According to the conservation of linear momentum, the total momentum after the collision should also be 6.0 kg⋅m/s. Let's break the final momentum into its x and y components, using the angles involved.
The x-component of the final momentum will be:
Final momentum x-component = (mass of the first ball * final velocity of the first ball in the x-direction) + (mass of the second ball * final velocity of the second ball in the x-direction)
Final momentum x-component = 2.0 kg * v1f * cos(30°) + 2.0 kg * v2f * cos(0°)
Final momentum x-component = 2.0 kg * v1f * (sqrt(3)/2) + 2.0 kg * v2f
The y-component of the final momentum will be:
Final momentum y-component = (mass of the first ball * final velocity of the first ball in the y-direction) + (mass of the second ball * final velocity of the second ball in the y-direction)
Final momentum y-component = 2.0 kg * v1f * sin(30°) + 2.0 kg * v2f * sin(0°)
Final momentum y-component = 2.0 kg * v1f * (1/2)
Since the total momentum after the collision is conserved, we can equate it to the initial momentum:
Final momentum x-component + Final momentum y-component = Initial momentum
2.0 kg * v1f * (sqrt(3)/2) + 2.0 kg * v2f + 2.0 kg * v1f * (1/2) = 6.0 kg⋅m/s
Now, we have two unknowns, v1f and v2f, but we have two equations based on conservation of momentum. We can solve these equations simultaneously to find the values of v1f and v2f.
Equation 1: 2.0 kg * v1f * (sqrt(3)/2) + 2.0 kg * v2f + 2.0 kg * v1f * (1/2) = 6.0 kg⋅m/s
Since the collision is elastic, we can use another principle called conservation of kinetic energy. It states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The initial kinetic energy of the system is given by:
Initial kinetic energy = (1/2) * (mass of the first ball * (initial velocity of the first ball)^2) + (1/2) * (mass of the second ball * (initial velocity of the second ball)^2)
Initial kinetic energy = (1/2) * (2.0 kg * (3.0 m/s)^2) + (1/2) * (2.0 kg * 0^2)
Initial kinetic energy = 9.0 J
The final kinetic energy of the system can be calculated as:
Final kinetic energy = (1/2) * (mass of the first ball * (final velocity of the first ball)^2) + (1/2) * (mass of the second ball * (final velocity of the second ball)^2)
Final kinetic energy = (1/2) * (2.0 kg * v1f^2) + (1/2) * (2.0 kg * v2f^2)
Final kinetic energy = 2.0 kg * (v1f^2 + v2f^2)
Since the collision is elastic, the total initial kinetic energy should be equal to the total final kinetic energy.
Initial kinetic energy = Final kinetic energy
9.0 J = 2.0 kg * (v1f^2 + v2f^2)
Now, we have two equations based on conservation of kinetic energy and two equations based on conservation of momentum. We can solve these four equations simultaneously to find the values of v1f and v2f.