A boy on a 2.4 kg skateboard initially at rest
tosses a(n) 8.2 kg jug of water in the forward
direction.
If the jug has a speed of 3.0 m/s relative to
the ground and the boy and skateboard move
in the opposite direction at 0.57 m/s, find the
boy’s mass.
Answer in units of kg
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To find the boy's mass, we can use the principle of conservation of momentum. The total momentum before the jug is thrown is equal to the total momentum after the jug is thrown.
The momentum of an object is defined as the product of its mass and velocity. We can express the momentum using the formula:
Momentum = mass × velocity
Let's denote the mass of the boy as "m" (in kg) and the velocity of the boy and skateboard as "v" (in m/s).
Before the jug is thrown:
The momentum of the boy and skateboard is equal to the product of their combined mass (m + 2.4 kg) and their velocity (-0.57 m/s), since they move in the opposite direction. So, the momentum before the jug is thrown is:
Initial momentum = (m + 2.4 kg) × (-0.57 m/s)
After the jug is thrown:
The momentum of the boy and skateboard is equal to the product of their combined mass (m + 2.4 kg) and their new velocity (v - 0.57 m/s, since 0.57 m/s is subtracted from the velocity). Additionally, the momentum of the jug is the product of its mass (8.2 kg) and its velocity (3.0 m/s). So, the momentum after the jug is thrown is:
Final momentum = (m + 2.4 kg) × (v - 0.57 m/s) + (8.2 kg × 3.0 m/s)
According to the conservation of momentum, the initial momentum should be equal to the final momentum. Therefore, we can write the equation:
(m + 2.4 kg) × (-0.57 m/s) = (m + 2.4 kg) × (v - 0.57 m/s) + (8.2 kg × 3.0 m/s)
Now we can solve this equation for "m," which will give us the boy's mass.
Let's simplify the equation step by step:
Step 1: Distribute the terms on the right side of the equation:
(-0.57 m/s) × (m + 2.4 kg) = (v - 0.57 m/s) × (m + 2.4 kg) + (8.2 kg × 3.0 m/s)
Step 2: Expand the terms on both sides:
-0.57 m/s × m - 0.57 m/s × 2.4 kg = v × m + v × 2.4 kg - 0.57 m/s × m - 0.57 m/s × 2.4 kg + 8.2 kg × 3.0 m/s
Step 3: Combine like terms:
-0.57 m + (-1.368 kgm/s) = vm + (2.4v - 1.368 kgm/s) + (24.6 kgm/s)
Step 4: Simplify further:
-0.57 m - 1.368 kgm/s = vm + 2.4v - 1.368 kgm/s + 24.6 kgm/s
Step 5: Cancel out similar terms:
-0.57 m - 1.368 kgm/s = vm + 2.4v + 24.6 kgm/s
Step 6: Rearrange the terms:
-0.57 m - vm = 2.4v + 24.6 kgm/s - 1.368 kgm/s
Step 7: Combine the constant terms:
-0.57 m - vm = 2.4v + (24.6 kgm/s - 1.368 kgm/s)
Step 8: Simplify further:
-0.57 m - vm = 2.4v + 23.232 kgm/s
Step 9: Move the terms involving "m" to one side and those involving "v" to the other side:
-0.57 m - vm - 2.4v = 23.232 kgm/s
Step 10: Combine like terms:
(-0.57 m - vm) - 2.4v = 23.232 kgm/s
Step 11: Factor out "m":
m(-0.57 - v) = 23.232 kgm/s + 2.4v
Step 12: Divide both sides by (-0.57 - v):
m = (23.232 kgm/s + 2.4v) / (-0.57 - v)
Now, plug in the given values for "v" (3.0 m/s) and solve for "m":
m = (23.232 kgm/s + 2.4 × 3.0 m/s) / (-0.57 - 3.0 m/s)
Evaluate the expression on the right side to get the value of "m."