A billiard ball moving at 6.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 5.39 m/s, at an angle of 26.0 with respect to the original line of motion
(a) Find the velocity (magnitude and direction) of the second ball after collision
initial x momentum = 6 m
initial y momentum = 0
final x momentum = m(5.39 cos 26) + m Vx
final y momentum = m (5.39 sin 26)+ m Vy
m cancels
initial = final
solve for Vx and Vy
To find the velocity of the second ball after the collision, we need to use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
The momentum of an object is given by the product of its mass and velocity. Let's denote the mass of each ball as "m". Given that the first ball has a velocity of 6.00 m/s and moves at an angle of 26.0° with respect to the original line of motion, we can break down its velocity into two components:
Vx = 6.00 m/s * cos(26.0°)
Vy = 6.00 m/s * sin(26.0°)
The momentum of the first ball before the collision is given by:
P_initial = mv_initial = m * 6.00 m/s
After the collision, the first ball moves at a velocity of 5.39 m/s, making an angle of 26.0° with respect to the original line of motion. Breaking down its velocity into components:
Vx_final = 5.39 m/s * cos(26.0°)
Vy_final = 5.39 m/s * sin(26.0°)
The momentum of the second ball after the collision is given by:
P_final = mv_final = m * V_final
According to the conservation of momentum, the total initial momentum of the system (before the collision) must be equal to the total final momentum (after the collision):
P_initial = P_final
m * 6.00 m/s = m * V_final
Canceling out the mass:
6.00 m/s = V_final
Therefore, the magnitude of the velocity of the second ball after the collision is 6.00 m/s. The direction of its velocity is not specified in the given information, so we cannot determine the exact angle or direction of the second ball after the collision.
To find the velocity (magnitude and direction) of the second ball after the collision, we can use the principle of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision. The momentum of an object is given by the product of its mass and velocity.
Let's assume the mass of each ball is "m".
Before the collision:
The first ball has a velocity (magnitude) of 6.00 m/s. Its momentum is given by: P1 = mass * velocity = m * 6.00.
Since the second ball is stationary, its velocity (magnitude) is 0. Therefore, its momentum is also 0: P2 = mass * velocity = m * 0 = 0.
After the collision:
The first ball has a velocity (magnitude) of 5.39 m/s at an angle of 26.0 degrees with respect to the original line of motion. Its momentum is given by: P1' = mass * velocity = m * 5.39.
The second ball has a velocity (magnitude) v2 and an unknown direction. Its momentum is given by: P2' = mass * velocity = m * v2.
According to the principle of conservation of momentum, P1 + P2 = P1' + P2'.
Therefore, we have: m * 6.00 + 0 = m * 5.39 + m * v2.
Simplifying the equation: 6.00 = 5.39 + v2.
Now, we can solve for v2:
v2 = 6.00 - 5.39 = 0.61 m/s.
Therefore, the velocity (magnitude) of the second ball after the collision is 0.61 m/s.
To find the direction, we need to determine the angle it makes with respect to the original line of motion. This can be determined using trigonometry.
tanθ = (opposite / adjacent)
tanθ = (5.39 * sin(26.0)) / (5.39 * cos(26.0))
tanθ = sin(26.0) / cos(26.0)
θ = arctan(tanθ)
θ = arctan(sin(26.0) / cos(26.0))
Using the inverse tangent function, we can determine the angle θ to be approximately 26.0 degrees.
Therefore, the velocity (magnitude and direction) of the second ball after the collision is 0.61 m/s at an angle of approximately 26.0 degrees with respect to the original line of motion.