Ann, a 59.2 kg person throws a 0.0435 kg snowball forward with a ground speed of 39.5 m/s. Billy, with a mass of 63.5 kg, catches the snowball. Both people are on skates. Ann is initially moving forward with a speed of 2.22 m/s, and Billy is initially at rest. After the snowball is exchanged (a) What is Ann’s velocity?
To find Ann's velocity after exchanging the snowball, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the exchange is equal to the total momentum after the exchange.
The initial momentum of Ann can be calculated as the product of her mass and her initial velocity:
Initial momentum of Ann = mass of Ann × initial velocity of Ann
Then, we need to calculate the initial momentum of the snowball, which is the product of the snowball's mass and Ann's initial velocity (since they are moving together initially):
Initial momentum of snowball = mass of snowball × initial velocity of Ann
The total initial momentum before the exchange is simply the sum of these two momenta:
Total initial momentum = Initial momentum of Ann + Initial momentum of snowball
We can also calculate the final momentum of Ann after the exchange by multiplying her mass and her final velocity:
Final momentum of Ann = mass of Ann × final velocity of Ann
Since the total momentum before the exchange is equal to the total momentum after the exchange, we can set up an equation:
Total initial momentum = Final momentum of Ann
Now we can substitute the values given in the problem into these equations and solve for the final velocity of Ann.
Given values:
Mass of Ann (m1) = 59.2 kg
Initial velocity of Ann (v1) = 2.22 m/s
Mass of snowball (m2) = 0.0435 kg
Initial velocity of snowball (v2) = 39.5 m/s
Using the equations mentioned above, we can now calculate the final velocity of Ann:
Initial momentum of Ann = m1 × v1
Initial momentum of snowball = m2 × v1 (since they have the same initial velocity)
Total initial momentum = Initial momentum of Ann + Initial momentum of snowball
Now we can solve for the total initial momentum:
Total initial momentum = (m1 × v1) + (m2 × v1)
Since the total initial momentum is equal to the final momentum of Ann:
Total initial momentum = Final momentum of Ann
We can now set up the equation and solve for the final velocity of Ann:
(m1 × v1) + (m2 × v1) = m1 × final velocity of Ann
Substituting the given values into this equation:
(59.2 kg × 2.22 m/s) + (0.0435 kg × 39.5 m/s) = 59.2 kg × final velocity of Ann
Now we can solve for the final velocity of Ann:
Final velocity of Ann = ((59.2 kg × 2.22 m/s) + (0.0435 kg × 39.5 m/s)) / 59.2 kg
Final velocity of Ann = (130.944 kg·m/s + 1.71725 kg·m/s) / 59.2 kg
Final velocity of Ann = 132.66125 kg·m/s / 59.2 kg
Final velocity of Ann ≈ 2.241 m/s
Therefore, after exchanging the snowball, Ann's velocity will be approximately 2.241 m/s.