Problem 3- A 10-g metal ball is moving to the left with a velocity of magnitude 0.4 m/s has a head-on, elastic collision with a larger 30-g metal ball moving to the right with a velocity of magnitude 0.2 m/s.
a) Find the speed of the two balls after the collision.
b) If the 10g ball is made of mud that stich with the 30g ball after the collision. What is the final speed of the taw ball, and the direction of motion?
If it is elastic, not only momentum is the same before and after but also kinetic energy
If they stick together, then only momentum is conserved but the single mass after collision is m1 + m2
part a
m1 Vi = m1 u + m2 v
(1/2)m1 Vi^2 = (1/2)m1 u^2 +(1/2)m2 v^2
part b
m1 Vi = (m1+m2) v
To solve this problem, we can use the principle of conservation of momentum and the equation for elastic collisions. Here's how you can calculate the answers to both parts of the problem:
a) Find the speed of the two balls after the collision:
1. Calculate the initial momentum of each ball before the collision:
- Ball 1 (10g) moving to the left: momentum = mass x velocity = 0.01 kg x (-0.4 m/s) = -0.004 kg*m/s
- Ball 2 (30g) moving to the right: momentum = mass x velocity = 0.03 kg x 0.2 m/s = 0.006 kg*m/s
2. Apply the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision:
- Total initial momentum = Total final momentum
- -0.004 kg*m/s + 0.006 kg*m/s = (mass of ball 1 + mass of ball 2) x final velocity
3. Rearrange the equation to solve for the final velocity:
- (mass of ball 1 + mass of ball 2) x final velocity = 0.006 kg*m/s - 0.004 kg*m/s
- (0.01 kg + 0.03 kg) x final velocity = 0.002 kg*m/s
- 0.04 kg x final velocity = 0.002 kg*m/s
- final velocity = 0.002 kg*m/s / 0.04 kg = 0.05 m/s
Therefore, the speed of both balls after the collision is 0.05 m/s.
b) If the 10g ball is made of mud that sticks with the 30g ball after the collision, the final speed of the combined ball can be calculated by treating it as a single object:
1. Find the total mass of the combined balls:
- Total mass = mass of ball 1 + mass of ball 2 = 0.01 kg + 0.03 kg = 0.04 kg
2. Calculate the momentum of the combined balls using the final velocity obtained in part a:
- Momentum = mass x velocity = 0.04 kg x 0.05 m/s = 0.002 kg*m/s
3. Since the two balls stick together, the final speed of the combined ball is equal to the momentum divided by the total mass:
- Final velocity = total momentum / total mass = 0.002 kg*m/s / 0.04 kg = 0.05 m/s
Therefore, the final speed of the combined ball (10g ball made of mud stuck to the 30g ball) is 0.05 m/s. The direction of motion is to the right, as it was the direction of the initial velocity of the 30g ball.