A billiard ball moving at 5.45 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.81 m/s, at an angle of 28° with respect to the original line of motion. Find the velocity (MAGNITUDE and DIRECTION) of the second ball after collision.
______ m/s
______ degrees
To find the velocity of the second ball after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
Momentum is conserved during a collision, which means that the total momentum before the collision is equal to the total momentum after the collision.
Let's assume the initial velocity of the second ball (stationary ball) is u₂ and the velocity of the first ball (moving ball) is u₁. The final velocities of the first and second balls after the collision are v₁ and v₂, respectively.
Using the conservation of momentum equation:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Since the second ball is initially stationary, its initial velocity (u₂) is 0 m/s. The masses of both balls are the same. Therefore, we have:
m₁u₁ = m₁v₁ + m₂v₂
2. Conservation of Kinetic Energy:
Let's assume the mass of both balls is m, and the kinetic energy of the first ball before and after the collision is KE₁ and KE₁', respectively. The kinetic energy of the second ball after the collision is KE₂.
Using the conservation of kinetic energy equation:
KE₁ + KE₂ = KE₁'
Since the collision is elastic (no energy loss), the kinetic energy is conserved. Therefore, we can write:
(1/2)m(u₁)² + 0 = (1/2)m(v₁)² + (1/2)m(v₂)²
Now, let's plug in the given values:
u₁ = 5.45 m/s (initial velocity of first ball)
v₁ = 4.81 m/s (final velocity of first ball)
θ = 28° (angle of the first ball with respect to the original line of motion)
To find the magnitude of the final velocity of the second ball (v₂):
1. Calculate the x-component of the final velocity of the first ball (v₁x) using the angle and the initial velocity of the first ball:
v₁x = v₁ * cos(θ)
2. Calculate the y-component of the final velocity of the first ball (v₁y) using the angle and the initial velocity of the first ball:
v₁y = v₁ * sin(θ)
3. Calculate the x-component of the final velocity of the second ball (v₂x) using the conservation of momentum equation:
v₂x = ((m₁ - m₂) * v₁x + 2 * m₂ * u₁) / (m₁ + m₂)
4. Calculate the y-component of the final velocity of the second ball (v₂y) using the conservation of momentum equation:
v₂y = v₁y
5. Calculate the magnitude of the final velocity of the second ball (v₂) using the Pythagorean theorem:
v₂ = √(v₂x² + v₂y²)
6. Calculate the angle of the final velocity of the second ball (θ₂) using the inverse tangent function:
θ₂ = atan(v₂y / v₂x)
Now, let's calculate the values:
1. Calculate the x-component of the final velocity of the first ball (v₁x):
v₁x = 4.81 m/s * cos(28°) = 4.205 m/s
2. Calculate the y-component of the final velocity of the first ball (v₁y):
v₁y = 4.81 m/s * sin(28°) = 2.154 m/s
3. Calculate the x-component of the final velocity of the second ball (v₂x):
v₂x = ((1 - 1) * 4.205 m/s + 2 * 0 m/s) / (1 + 1) = 0 m/s (since the stationary ball doesn't move in the x-direction)
4. Calculate the y-component of the final velocity of the second ball (v₂y):
v₂y = 2.154 m/s
5. Calculate the magnitude of the final velocity of the second ball (v₂):
v₂ = √(0 m/s)² + (2.154 m/s)² = 2.154 m/s
6. Calculate the angle of the final velocity of the second ball (θ₂):
θ₂ = atan(2.154 m/s / 0 m/s) = 90°
Therefore, the magnitude of the final velocity of the second ball after the collision is 2.154 m/s, and the direction is 90 degrees (or straight up).
______ m/s
______ degrees