A 100g golf ball moving with a velocity of 20 m/s, collides with an 8 kg steel ball at rest,if the collision is elastic,compute the velocities of both balls after the collision.
What is center_of mass velocity??
To compute the velocities of both balls after the collision, we can use the principle of conservation of momentum.
1. Start by calculating the initial momentum (p_initial) of the golf ball and the steel ball.
- The initial momentum of an object is the product of its mass and velocity.
- Momentum (p) = mass (m) × velocity (v)
The initial momentum of the golf ball can be calculated as:
Momentum_GolfBall_initial = mass_GolfBall × velocity_GolfBall
= 0.1 kg × 20 m/s
The initial momentum of the steel ball, since it is at rest, is zero.
2. Since the collision is elastic, the total momentum before the collision (p_initial) must be equal to the total momentum after the collision (p_final).
Momentum_GolfBall_initial + Momentum_SteelBall_initial = Momentum_GolfBall_final + Momentum_SteelBall_final
0.1 kg × 20 m/s + 0 kg = Mass_GolfBall × Velocity_GolfBall_final + Mass_SteelBall × Velocity_SteelBall_final
3. Now, we need to consider the conservation of kinetic energy because the collision is elastic. For an elastic collision:
- Both momentum and kinetic energy are conserved.
The kinetic energy (KE) is given by:
KE = (1/2) × mass × velocity^2
The initial kinetic energy (KE_initial) of the golf ball can be calculated as:
KE_GolfBall_initial = (1/2) × mass_GolfBall × velocity_GolfBall^2
The initial kinetic energy (KE_initial) of the steel ball is zero since it is at rest.
The total kinetic energy before the collision (KE_initial) must be equal to the total kinetic energy after the collision (KE_final).
KE_GolfBall_initial + KE_SteelBall_initial = KE_GolfBall_final + KE_SteelBall_final
(1/2) × mass_GolfBall × velocity_GolfBall^2 + 0 = (1/2) × Mass_GolfBall × Velocity_GolfBall_final^2 + (1/2) × Mass_SteelBall × Velocity_SteelBall_final^2
4. Now, we have two equations:
Equation 1: 0.1 kg × 20 m/s + 0 kg = Mass_GolfBall × Velocity_GolfBall_final + 8 kg × Velocity_SteelBall_final
Equation 2: (1/2) × 0.1 kg × (20 m/s)^2 + 0 = (1/2) × Mass_GolfBall × Velocity_GolfBall_final^2 + (1/2) × 8 kg × Velocity_SteelBall_final^2
5. By solving these two equations simultaneously, we can find the velocities of both balls after the collision.
1.5m/s
Both kinetic energy and momentum are conserved. Assume a head-on collision with motion along the x axis only.
There is an easy way to do this without aq lot of algebra.
The center-of mass velocity is
Vcm = 0.1*20/8.1 = 0.2469 m/s
In a coordinate system travelling at Vcm, both balls reverse direction and keep the same speed.
Golf ball initial velocity in CM system = 20 - 0.2469 = 19.753 m/s
Golf ball final velocity in CM system = -19.753 m/s
Golf ball final velocity in lab coordinates = -19.753 +.2469 = -19.506 m/s
Steel ball initial velocity in CM coordinates = -0.2469 m/s
Steel ball final velocity in CM coordinates = +0.2469 m/s
Steel ball final velocity in lab coordinates = +0.4938 m/s
Check: Final KE = 0.98 + 19.02 = 20.00 J = Initial KE
Final momentum = 3.9504 - 1.9506 = 2.00 kg m/s = Initial momentum