A 69.0 kg person throws a 0.0410 kg snowball forward with a ground speed of 35.0 m/s. A second person, with a mass of 60.0 kg, catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of 2.00 m/s, and the second person is initially at rest. What are the velocities of the two people after the snowball is exchanged? Disregard the friction between the skates and the ice.
To solve this problem, we can apply the principle of conservation of momentum. The total momentum before the snowball is thrown is equal to the total momentum after the snowball is exchanged.
Let's define the variables:
m1 = mass of the first person (69.0 kg)
v1i = initial velocity of the first person (2.00 m/s)
m2 = mass of the second person (60.0 kg)
v2i = initial velocity of the second person (0 m/s)
m_snowball = mass of the snowball (0.0410 kg)
v_snowball = velocity of the snowball (35.0 m/s)
v1f = final velocity of the first person
v2f = final velocity of the second person
Now, we can calculate the initial momentum before the snowball is thrown:
Initial momentum of person 1 = m1 * v1i
Initial momentum of person 2 = m2 * v2i
Initial momentum of snowball = m_snowball * v_snowball
The total initial momentum is the sum of the individual momenta:
Initial total momentum = (m1 * v1i) + (m2 * v2i) + (m_snowball * v_snowball)
The total final momentum is equal to the total initial momentum because of the conservation of momentum:
Final total momentum = (m1 * v1f) + (m2 * v2f) + (m_snowball * v_snowball)
Setting the initial and final total momentum equal:
(m1 * v1i) + (m2 * v2i) + (m_snowball * v_snowball) = (m1 * v1f) + (m2 * v2f) + (m_snowball * v_snowball)
We can now solve this equation for v1f and v2f.
First, rewrite the equation:
(m1 * v1i) + (m2 * v2i) = (m1 * v1f) + (m2 * v2f)
Rearrange the equation to isolate v1f:
m1 * v1f = (m1 * v1i) + (m2 * v2i) - (m2 * v2f)
Divide by m1:
v1f = (m1 * v1i) / m1 + (m2 * v2i) / m1 - (m2 * v2f) / m1
Simplify:
v1f = v1i + (m2 * v2i) / m1 - (m2 * v2f) / m1
Now, substitute the given values into the equation and solve for v1f:
v1f = 2.00 m/s + (60.0 kg * 0 m/s) / 69.0 kg - (60.0 kg * v2f) / 69.0 kg
Next, rearrange the equation to isolate v2f:
(m2 * v2f) / m1 = (m2 * v2i) / m1 + (m1 * v1i) / m1 - v1f
Multiply by m1 to get rid of the denominator:
m2 * v2f = m2 * v2i + m1 * v1i - v1f * m1
Divide by m2:
v2f = (m2 * v2i + m1 * v1i - v1f * m1) / m2
Now, substitute the given values into the equation and solve for v2f:
v2f = (60.0 kg * 0 m/s + 69.0 kg * 2.00 m/s - 69.0 kg * v1f) / 60.0 kg
By plugging in the values and solving the equations, we can find the final velocities of the two people after the snowball is exchanged.