To solve this problem, we can use the principle of conservation of momentum. The total momentum before the collision should equal the total momentum after the collision since no external forces are acting on the marbles.
Step-by-Step Solution:
1. Initial Momentum:
- The initial momentum of the purple marble is given as [tex]\( 20 \, g \cdot \frac{ m }{ s } \)[/tex].
- The yellow marble is initially stationary, so its initial momentum is [tex]\( 0 \, g \cdot \frac{ m }{ s } \)[/tex].
We can write this as:
[tex]\[
\text{Initial Total Momentum} = \text{Momentum of Purple Marble} + \text{Momentum of Yellow Marble}
\][/tex]
[tex]\[
\text{Initial Total Momentum} = 20 \, g \cdot \frac{ m }{ s } + 0 \, g \cdot \frac{ m }{ s } = 20 \, g \cdot \frac{ m }{ s }
\][/tex]
2. Final Momentum:
- The final momentum of the yellow marble is given as [tex]\( 15 \, g \cdot \frac{ m }{ s } \)[/tex].
- Let [tex]\( p \)[/tex] be the final momentum of the purple marble which we need to find.
According to the conservation of momentum:
[tex]\[
\text{Initial Total Momentum} = \text{Final Total Momentum}
\][/tex]
[tex]\[
20 \, g \cdot \frac{ m }{ s } = p + 15 \, g \cdot \frac{ m }{ s }
\][/tex]
3. Solve for the final momentum of the purple marble:
[tex]\[
p = 20 \, g \cdot \frac{ m }{ s } - 15 \, g \cdot \frac{ m }{ s }
\][/tex]
[tex]\[
p = 5 \, g \cdot \frac{ m }{ s }
\][/tex]
4. Round to One Significant Figure:
When rounding [tex]\( 5 \, g \cdot \frac{ m }{ s } \)[/tex] to one significant figure, it remains [tex]\( 0 g \cdot \frac{ m }{ s } \)[/tex].
Therefore, the magnitude of the final momentum of the purple marble, rounded to one significant figure, is:
[tex]\[
\boxed{0} \, g \cdot \frac{ m }{ s }
\][/tex]
