Answer :
Let's break down the solution in a detailed, step-by-step manner to determine the comparisons correctly given the masses and the fact that all balls have the same momentum.
First, recall that the momentum [tex]\( p \)[/tex] is given by the formula:
[tex]\[ p = m \cdot v \][/tex]
Where:
- [tex]\( p \)[/tex] is the momentum
- [tex]\( m \)[/tex] is the mass
- [tex]\( v \)[/tex] is the velocity
Since momentum [tex]\( p \)[/tex] is the same for all the balls, we can write for any ball:
[tex]\[ p = m_{\text{green}} \cdot v_{\text{green}} = m_{\text{red}} \cdot v_{\text{red}} = m_{\text{yellow}} \cdot v_{\text{yellow}} = m_{\text{purple}} \cdot v_{\text{purple}} \][/tex]
To find the velocities, we rearrange this formula to solve for [tex]\( v \)[/tex]:
[tex]\[ v = \frac{p}{m} \][/tex]
So, the velocity of each ball is inversely proportional to its mass:
- Green ball: [tex]\( v_{\text{green}} = \frac{p}{0.5} \)[/tex]
- Red ball: [tex]\( v_{\text{red}} = \frac{p}{1.2} \)[/tex]
- Yellow ball: [tex]\( v_{\text{yellow}} = \frac{p}{0.9} \)[/tex]
- Purple ball: [tex]\( v_{\text{purple}} = \frac{p}{1.7} \)[/tex]
Due to the inverse proportionality, to simplify comparisons we can use the inverse of the masses:
- Inverse of Green ball's mass: [tex]\( \frac{1}{0.5} = 2.0 \)[/tex]
- Inverse of Red ball's mass: [tex]\( \frac{1}{1.2} \approx 0.833 \)[/tex]
- Inverse of Yellow ball's mass: [tex]\( \frac{1}{0.9} \approx 1.111 \)[/tex]
- Inverse of Purple ball's mass: [tex]\( \frac{1}{1.7} \approx 0.588 \)[/tex]
Now we analyze the statements based on these inverse values, noting that a higher inverse mass corresponds to a higher velocity:
1. The green ball has a lower velocity than the purple ball.
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
- Inverse velocity of Purple ball: [tex]\( 0.588 \)[/tex]
Since [tex]\( 2.0 > 0.588 \)[/tex], the green ball actually has a higher velocity than the purple ball. Therefore, this statement is false.
2. The red ball has a greater velocity than the purple ball.
- Inverse velocity of Red ball: [tex]\( 0.833 \)[/tex]
- Inverse velocity of Purple ball: [tex]\( 0.588 \)[/tex]
Since [tex]\( 0.833 > 0.588 \)[/tex], the red ball has a greater velocity than the purple ball. Therefore, this statement is true.
3. The yellow ball has a higher velocity than the green ball.
- Inverse velocity of Yellow ball: [tex]\( 1.111 \)[/tex]
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
Since [tex]\( 1.111 < 2.0 \)[/tex], the yellow ball actually has a lower velocity than the green ball. Therefore, this statement is false.
4. The red ball has a greater velocity than the green ball.
- Inverse velocity of Red ball: [tex]\( 0.833 \)[/tex]
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
Since [tex]\( 0.833 < 2.0 \)[/tex], the red ball has a lower velocity than the green ball. Therefore, this statement is false.
Thus, the correct comparison between the velocities of two balls is:
The red ball has a greater velocity than the purple ball.
First, recall that the momentum [tex]\( p \)[/tex] is given by the formula:
[tex]\[ p = m \cdot v \][/tex]
Where:
- [tex]\( p \)[/tex] is the momentum
- [tex]\( m \)[/tex] is the mass
- [tex]\( v \)[/tex] is the velocity
Since momentum [tex]\( p \)[/tex] is the same for all the balls, we can write for any ball:
[tex]\[ p = m_{\text{green}} \cdot v_{\text{green}} = m_{\text{red}} \cdot v_{\text{red}} = m_{\text{yellow}} \cdot v_{\text{yellow}} = m_{\text{purple}} \cdot v_{\text{purple}} \][/tex]
To find the velocities, we rearrange this formula to solve for [tex]\( v \)[/tex]:
[tex]\[ v = \frac{p}{m} \][/tex]
So, the velocity of each ball is inversely proportional to its mass:
- Green ball: [tex]\( v_{\text{green}} = \frac{p}{0.5} \)[/tex]
- Red ball: [tex]\( v_{\text{red}} = \frac{p}{1.2} \)[/tex]
- Yellow ball: [tex]\( v_{\text{yellow}} = \frac{p}{0.9} \)[/tex]
- Purple ball: [tex]\( v_{\text{purple}} = \frac{p}{1.7} \)[/tex]
Due to the inverse proportionality, to simplify comparisons we can use the inverse of the masses:
- Inverse of Green ball's mass: [tex]\( \frac{1}{0.5} = 2.0 \)[/tex]
- Inverse of Red ball's mass: [tex]\( \frac{1}{1.2} \approx 0.833 \)[/tex]
- Inverse of Yellow ball's mass: [tex]\( \frac{1}{0.9} \approx 1.111 \)[/tex]
- Inverse of Purple ball's mass: [tex]\( \frac{1}{1.7} \approx 0.588 \)[/tex]
Now we analyze the statements based on these inverse values, noting that a higher inverse mass corresponds to a higher velocity:
1. The green ball has a lower velocity than the purple ball.
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
- Inverse velocity of Purple ball: [tex]\( 0.588 \)[/tex]
Since [tex]\( 2.0 > 0.588 \)[/tex], the green ball actually has a higher velocity than the purple ball. Therefore, this statement is false.
2. The red ball has a greater velocity than the purple ball.
- Inverse velocity of Red ball: [tex]\( 0.833 \)[/tex]
- Inverse velocity of Purple ball: [tex]\( 0.588 \)[/tex]
Since [tex]\( 0.833 > 0.588 \)[/tex], the red ball has a greater velocity than the purple ball. Therefore, this statement is true.
3. The yellow ball has a higher velocity than the green ball.
- Inverse velocity of Yellow ball: [tex]\( 1.111 \)[/tex]
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
Since [tex]\( 1.111 < 2.0 \)[/tex], the yellow ball actually has a lower velocity than the green ball. Therefore, this statement is false.
4. The red ball has a greater velocity than the green ball.
- Inverse velocity of Red ball: [tex]\( 0.833 \)[/tex]
- Inverse velocity of Green ball: [tex]\( 2.0 \)[/tex]
Since [tex]\( 0.833 < 2.0 \)[/tex], the red ball has a lower velocity than the green ball. Therefore, this statement is false.
Thus, the correct comparison between the velocities of two balls is:
The red ball has a greater velocity than the purple ball.
