A 55-kg woman contestant on a reality television show is at rest at the south end of a horizontal 148-kg raft that is floating in crocodile-infested waters. She and the raft are initially at rest. She needs to jump from the raft to a platform that is several meters off the north end of the raft. She takes a running start. When she reaches the north end of the raft she is running at 4.4 m/s relative to the raft. At that instant, what is her velocity relative to the water?
I did this....
m(w)*v(w) + m(r)*v(r) = 0
v(r) = -1.64
v(w) + v(r) = v(w)(relative to water) = 2.76 = incorrect?
To answer this question, we can use the principle of conservation of momentum. The total momentum before the woman jumps is equal to the total momentum after she jumps.
Let's define the variables:
m(w) = mass of the woman = 55 kg
v(w) = velocity of the woman relative to the water (what we're trying to find)
m(r) = mass of the raft = 148 kg
v(r) = velocity of the raft relative to the water
The total momentum before the woman jumps is zero since both the woman and the raft are at rest:
m(w) * 0 + m(r) * 0 = 0
After the woman jumps, the total momentum must still be zero:
m(w) * v(w) + m(r) * v(r) = 0
We are given that the woman's velocity relative to the raft at the north end is 4.4 m/s. Let's plug in the given values:
55 kg * 4.4 m/s + 148 kg * v(r) = 0
Now, solve for v(r):
v(r) = - (55 kg * 4.4 m/s) / 148 kg
v(r) ≈ -1.639 m/s
The negative sign indicates that the direction is opposite to the initial direction of the woman's velocity. This means the raft is moving in the opposite direction to the north, i.e., toward the south.
To find the woman's velocity relative to the water, we can simply add the velocity of the raft to the woman's velocity relative to the raft:
v(w) = v(r) + 4.4 m/s
v(w) ≈ -1.639 m/s + 4.4 m/s
v(w) ≈ 2.761 m/s
Therefore, the woman's velocity relative to the water is approximately 2.761 m/s, not 2.76 m/s as you calculated.
Your equation m(w)*v(w) + m(r)*v(r) = 0 is incorrect. The correct equation to use in this situation is the conservation of momentum equation:
m(w)*v(w) + m(r)*v(r) = (m(w)+m(r))*v(final)
where:
m(w) is the mass of the woman (55 kg)
v(w) is the velocity of the woman relative to the raft (4.4 m/s)
m(r) is the mass of the raft (148 kg)
v(r) is the velocity of the raft relative to the water (unknown)
v(final) is the final velocity of the system (woman + raft) relative to the water (unknown)
Substituting the given values into the equation:
(55 kg)*(4.4 m/s) + (148 kg)*(v(r)) = (55 kg + 148 kg)*(v(final))
(242 kg*m/s) + (148 kg)*(v(r)) = (203 kg)*(v(final))
Now, you need to use the fact that the initial system is at rest, so the initial velocity of the entire system (woman + raft) relative to the water is zero:
(242 kg*m/s) + (148 kg)*(v(r)) = 0
Now, solve for v(r):
(148 kg)*(v(r)) = - (242 kg*m/s)
v(r) = - (242 kg*m/s) / (148 kg)
Using a calculator, you'll find:
v(r) = -1.64 m/s (approx)
Therefore, the woman's velocity relative to the water is approximately -1.64 m/s.