A boy throws a 4-kg pumpkin at 8 m/s to a 40-kg girl on roller skates, who catches it. At what speed does the girl then move backward?
Well, it sounds like the girl was caught off-guard! Let's see how fast she ends up moving backward after catching that pumpkin. To figure that out, we need to use the law of conservation of momentum.
The momentum of an object is equal to its mass multiplied by its velocity. Initially, the boy's pumpkin had a momentum of (4 kg) × (8 m/s) = 32 kg•m/s.
When the girl catches the pumpkin, she gains that momentum and starts moving backward. Since momentum is conserved, the total momentum after the catch must still be 32 kg•m/s.
Since the girl's mass is 40 kg, we can rearrange the equation to solve for her velocity:
Girl's momentum = Girl's mass × Girl's velocity
32 kg•m/s = 40 kg × Girl's velocity
Dividing both sides by 40 kg:
Girl's velocity = 32 kg•m/s ÷ 40 kg
This gives us a velocity of 0.8 m/s. So after catching the pumpkin, the girl will be moving backward at a speed of 0.8 m/s.
To solve this problem, we'll need to apply the principle of conservation of momentum. According to this principle, the total momentum before the pumpkin is thrown should be equal to the total momentum after the girl catches it.
Let's assign some variables:
Mass of the pumpkin, m1 = 4 kg
Initial velocity of the pumpkin, u1 = 8 m/s
Mass of the girl, m2 = 40 kg
Final velocity of the pumpkin and the girl, v2 = ? (to be determined)
Using the conservation of momentum equation:
m1 * u1 = (m1 + m2) * v2
Substituting the given values:
4 kg * 8 m/s = (4 kg + 40 kg) * v2
32 kg * m/s = 44 kg * v2
Now we can solve for v2 by dividing both sides of the equation by 44 kg:
32 kg * m/s / 44 kg = v2
v2 = 0.727 m/s
Therefore, the girl will move backward with a velocity of 0.727 m/s.
To determine the speed at which the girl moves backward after catching the pumpkin, we need to apply the conservation of momentum principle. According to this principle, the total momentum before the girl catches the pumpkin should be equal to the total momentum after she catches it.
Before the girl catches the pumpkin, the total momentum is given by the sum of the boy's momentum and the pumpkin's momentum. The momentum of an object is calculated by multiplying its mass by its velocity.
The boy's momentum before he throws the pumpkin can be calculated as:
Momentum of the boy = mass of the boy × velocity of the boy
Given that the boy's mass is not provided, we can assume a mass of 50 kg as a reasonable approximation. Therefore:
Momentum of the boy = 50 kg × 8 m/s
Next, we need to calculate the pumpkin's initial momentum. The momentum of the pumpkin can be determined using the same formula:
Momentum of the pumpkin = mass of the pumpkin × velocity of the pumpkin
Given that the mass of the pumpkin is 4 kg and the velocity of the pumpkin is 8 m/s, we can calculate its momentum:
Momentum of the pumpkin = 4 kg × 8 m/s
Now, the total momentum before the girl catches the pumpkin is the sum of the boy's momentum and the pumpkin's momentum:
Total momentum before = Momentum of the boy + Momentum of the pumpkin
Thus:
Total momentum before = (50 kg × 8 m/s) + (4 kg × 8 m/s)
To find the speed at which the girl moves backward after catching the pumpkin, we equate the total momentum before to the total momentum after. Since the girl is stationary before she catches the pumpkin, her initial momentum is zero. Afterward, the girl moves backward, so her velocity is in the opposite direction to the initial movement of the pumpkin. Let's assume the girl's mass is 40 kg.
The girl's momentum after catching the pumpkin can be obtained using the formula:
Momentum of the girl = mass of the girl × velocity of the girl
Since the girl moves backward, her velocity is negative, denoted by "-𝑣":
Momentum of the girl = 40 kg × (-𝑣)
According to the conservation of momentum principle, the total momentum before should be equal to the total momentum after:
Total momentum before = Total momentum after
Equating the two expressions, we have:
(50 kg × 8 m/s) + (4 kg × 8 m/s) = 40 kg × (-𝑣)
Simplifying the equation, we can solve for the velocity of the girl:
(400 kg⋅m/s) + (32 kg⋅m/s) = -40𝑣
432 kg⋅m/s = -40𝑣
𝑣 = (432 kg⋅m/s) / (-40 kg)
Therefore, the speed at which the girl moves backward after catching the pumpkin is:
𝑣 = -10.8 m/s
Note: The negative sign indicates that the girl's velocity is in the opposite direction to the initial movement of the pumpkin thrown by the boy.