A billiards ball of mass 0.4kg strikes another, initially stationary ball of the same mass. The first ball is deflected and travels at an angle of 30∘ with its original direction. Its initial speed was 12m/s and after the collision, it is 8m/s. What are the magnitude and direction of velocity of the second ball after collision?
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the mass of the object multiplied by its velocity: p = mv.
Let's denote the initial velocity of the first ball as v1i, the final velocity of the first ball after the collision as v1f, the initial velocity of the second ball as v2i, and the final velocity of the second ball after the collision as v2f.
The initial momentum can be calculated as:
p_initial = m1 * v1i + m2 * v2i
The final momentum can be calculated as:
p_final = m1 * v1f + m2 * v2f
Since the second ball is initially stationary, its initial velocity is zero: v2i = 0.
According to the conservation of momentum, p_initial = p_final. Thus, we can rewrite the equation as:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Plug in the values we know:
m1 = 0.4 kg (mass of the first ball)
m2 = 0.4 kg (mass of the second ball)
v1i = 12 m/s (initial velocity of the first ball)
v2i = 0 m/s (initial velocity of the second ball)
v1f = 8 m/s (final velocity of the first ball)
v2f = ? (final velocity of the second ball)
0.4 kg * 12 m/s + 0.4 kg * 0 m/s = 0.4 kg * 8 m/s + 0.4 kg * v2f
4.8 kg·m/s = 3.2 kg·m/s + 0.4 kg·v2f
v2f = (4.8 kg·m/s - 3.2 kg·m/s) / 0.4 kg
v2f = 12 m/s
Therefore, the final velocity of the second ball after the collision is 12 m/s.
2. Conservation of Kinetic Energy:
According to the principle of conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object is given by the formula: KE = (1/2) * m * v^2.
Let's denote the initial kinetic energy of the first ball as KE1i, the final kinetic energy of the first ball after the collision as KE1f, the initial kinetic energy of the second ball as KE2i, and the final kinetic energy of the second ball after the collision as KE2f.
The initial kinetic energy can be calculated as:
KE_initial = KE1i + KE2i
KE_initial = (1/2) * m1 * (v1i)^2 + (1/2) * m2 * (v2i)^2
The final kinetic energy can be calculated as:
KE_final = KE1f + KE2f
KE_final = (1/2) * m1 * (v1f)^2 + (1/2) * m2 * (v2f)^2
Since the second ball is initially stationary, its initial kinetic energy is zero: KE2i = 0.
According to the conservation of kinetic energy, KE_initial = KE_final. Thus, we can rewrite the equation as:
(1/2) * m1 * (v1i)^2 = (1/2) * m1 * (v1f)^2 + (1/2) * m2 * (v2f)^2
Plug in the values we know:
m1 = 0.4 kg (mass of the first ball)
m2 = 0.4 kg (mass of the second ball)
v1i = 12 m/s (initial velocity of the first ball)
v1f = 8 m/s (final velocity of the first ball)
v2f = 12 m/s (final velocity of the second ball)
(1/2) * 0.4 kg * (12 m/s)^2 = (1/2) * 0.4 kg * (8 m/s)^2 + (1/2) * 0.4 kg * (v2f)^2
34.56 J = 12.8 J + 0.2 kg * (v2f)^2
21.76 J = 0.2 kg * (v2f)^2
(v2f)^2 = 108.8 m^2/s^2
v2f = √(108.8 m^2/s^2)
v2f ≈ 10.44 m/s
Therefore, the magnitude of the velocity of the second ball after the collision is approximately 10.44 m/s.
To determine the direction, we can use the angle mentioned in the question (30 degrees). Since we know the magnitude and direction, we can represent it as a vector: 10.44 m/s at an angle of 30 degrees.
To find the magnitude and direction of velocity of the second ball after the collision, we can use the principle of conservation of linear momentum.
According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the initial velocity of the second ball as v2i and the final velocity as v2f.
The initial momentum of the system can be calculated as:
P_initial = m1 * v1i + m2 * v2i
The final momentum of the system can be calculated as:
P_final = m1 * v1f + m2 * v2f
Given:
m1 = m2 = 0.4 kg (mass of each ball)
v1i = 12 m/s (initial velocity of first ball)
v1f = 8 m/s (final velocity of first ball)
From the conservation of momentum principle, we can equate the initial and final momenta:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Substituting the known values:
0.4 * 12 + 0.4 * v2i = 0.4 * 8 + 0.4 * v2f
Simplifying the equation:
4.8 + 0.4 * v2i = 3.2 + 0.4 * v2f
0.4 * v2i = 3.2 - 4.8 + 0.4 * v2f
0.4 * v2i = -1.6 + 0.4 * v2f
Dividing both sides by 0.4:
v2i = -4 + v2f
Rearranging the equation:
v2f = v2i + 4
Since the second ball is initially stationary (v2i = 0), we can determine the magnitude and direction of the velocity of the second ball after the collision:
v2f = 0 + 4 = 4 m/s
Thus, the magnitude of the velocity of the second ball after the collision is 4 m/s. However, we do not have enough information to determine its direction.
conservation of momentum:
set up the equation in x,y directions, so you have two conservation of momentum equations. Notice two unknowns, direction and velocity of second ball. That should do it, two unknowns, two equations, and a bit of algebra (and basic trig). Here is a similar problem: https://www.jiskha.com/display.cgi?id=1391985689