A 2.00 kg blob of Gak, moving at 4.0 m/s, strikes a motionless figure skater in the back and sticks. The skater, with the Gak on her back, glides off at 0.160 m/s. Calculate the skater’s mass.
To solve this problem, we can apply the principle of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision. Mathematically, this can be stated as:
(mass of Gak * initial velocity of Gak) = (mass of skater * final velocity of skater)
Let's assign the following variables:
m1 = mass of Gak (2.00 kg)
v1 = initial velocity of Gak (4.0 m/s)
m2 = mass of skater (unknown)
v2 = final velocity of skater (0.160 m/s)
The equation can be written as:
(m1 * v1) = (m2 * v2)
Substituting the given values:
(2.00 kg * 4.0 m/s) = (m2 * 0.160 m/s)
Simplifying further:
8.00 kg·m/s = 0.16 m2·kg/s
Dividing both sides of the equation by 0.16 m/s:
50 kg = m2
Therefore, the skater's mass is 50 kg.
To calculate the skater's mass, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.
In this case, the system consists of the blob of Gak and the skater. Before the collision, the blob of Gak is moving at 4.0 m/s, and the skater is at rest. After the collision, the blob of Gak and the skater move together at 0.160 m/s.
First, let's calculate the momentum of the blob of Gak before the collision:
Momentum of the Gak before collision = mass of the Gak * velocity of the Gak before collision
Momentum of the Gak before collision = 2.00 kg * 4.0 m/s
Next, let's calculate the momentum of the skater and the Gak after the collision:
Momentum of the skater and Gak after collision = (mass of the skater + mass of the Gak) * velocity of the skater and Gak after collision
Momentum of the skater and Gak after collision = (mass of the skater + 2.00 kg) * 0.160 m/s
According to the principle of conservation of momentum, the momentum before the collision is equal to the momentum after the collision:
2.00 kg * 4.0 m/s = (mass of the skater + 2.00 kg) * 0.160 m/s
Now we can solve this equation to find the mass of the skater.
First, distribute the 0.160 m/s to (mass of the skater + 2.00 kg):
2.00 kg * 4.0 m/s = 0.160 m/s * mass of the skater + 0.160 m/s * 2.00 kg
Simplifying:
8 kg m/s = 0.32 m/s kg + 0.32 kg m/s
Combine the terms with the same units:
8 kg m/s = (0.32 kg + 0.32 kg) m/s
Simplify further:
8 kg m/s = 0.64 kg m/s
Now divide both sides of the equation by 0.64 kg m/s:
8 kg m/s / 0.64 kg m/s = mass of the skater
Finally, calculate the mass of the skater:
mass of the skater = 8 kg m/s / 0.64 kg m/s
mass of the skater ≈ 12.5 kg
Therefore, the mass of the skater is approximately 12.5 kg.
Neglect friction (because of the ice skates)and assume conservation of momentum. That results in the equation
2.00*4.0 = (M + 2.00)*0.160
Solve for M, which will be in units of kg
mvi + mvi(2) = mvf + mvf(2) all energy is conserved
2(4) = (m + 2).16 (the masses are now combined)
8 = .32m
m = 25 kg
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