Note: Take East as the positive direction.
A(n) 86 kg fisherman jumps from a dock
into a 131 kg rowboat at rest on the West side
of the dock.
If the velocity of the fisherman is 5.2 m/s
to the West as he leaves the dock, what is the
final velocity of the fisherman and the boat?
We can solve this problem using the conservation of momentum principle. The momentum before the jump is equal to the momentum after the jump.
Initial momentum (before the jump):
Fisherman: mass (m1) = 86 kg, velocity (v1) = -5.2 m/s (west direction is negative)
Rowboat: mass (m2) = 131 kg, velocity (v2) = 0 m/s
Initial momentum (P_initial) = m1 * v1 + m2 * v2
After the jump, the fisherman and rowboat move together with a final velocity (V_final).
Final momentum (P_final) = (m1 + m2) * V_final
According to the conservation of momentum, P_initial = P_final, so:
m1 * v1 + m2 * v2 = (m1 + m2) * V_final
Now, solve for V_final:
V_final = (m1 * v1 + m2 * v2) / (m1 + m2)
V_final = (86 * -5.2 + 131 * 0) / (86 + 131)
V_final = (-446.72 + 0) / 217
V_final = -446.72 / 217
V_final ≈ -2.06 m/s
So, the final velocity of the fisherman and the boat is -2.06 m/s, which is approximately 2.06 m/s to the West.
To find the final velocity of the fisherman and the boat, we can use the principle of conservation of momentum. According to this principle, the total momentum before the jump should be equal to the total momentum after the jump.
The total momentum before the jump is given by the equation:
(mass of fisherman × velocity of fisherman before the jump) + (mass of boat × velocity of boat before the jump) = Total momentum before the jump
Let's substitute in the given values:
(86 kg × 5.2 m/s) + (131 kg × 0 m/s) = Total momentum before the jump
This simplifies to:
(446.2 kg·m/s) + 0 = Total momentum before the jump
Total momentum before the jump = 446.2 kg·m/s
Since no external forces act on the system after the jump, the total momentum after the jump must also be 446.2 kg·m/s.
The final velocity of the fisherman and the boat can be calculated using the equation:
(mass of fisherman × velocity of fisherman after the jump) + (mass of boat × velocity of boat after the jump) = Total momentum after the jump
Let's assume the final velocity of the fisherman and the boat is v.
(86 kg × v) + (131 kg × v) = 446.2 kg·m/s
Simplifying the equation:
(86 kg + 131 kg) × v = 446.2 kg·m/s
217 kg × v = 446.2 kg·m/s
Dividing both sides by 217 kg:
v = 446.2 kg·m/s / 217 kg
v = 2.06 m/s (approximately)
Therefore, the final velocity of the fisherman and the boat is approximately 2.06 m/s to the East.
To solve this question, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the interaction is equal to the total momentum after the interaction.
Given:
Mass of the fisherman (m1) = 86 kg
Mass of the rowboat (m2) = 131 kg
Initial velocity of the fisherman (v1_initial) = -5.2 m/s (to the West)
Initial velocity of the rowboat (v2_initial) = 0 m/s (at rest)
Final velocity of the fisherman (v1_final) = ?
Final velocity of the rowboat (v2_final) = ?
Step 1: Write down the conservation of linear momentum equation:
(m1 * v1_initial) + (m2 * v2_initial) = (m1 * v1_final) + (m2 * v2_final)
Step 2: Substituting the given values into the equation:
(86 kg * -5.2 m/s) + (131 kg * 0 m/s) = (86 kg * v1_final) + (131 kg * v2_final)
Step 3: Simplify the equation:
-447.2 kg*m/s = 86 kg * v1_final + 131 kg * v2_final
Step 4: Solve the equation for v1_final and v2_final:
v1_final = (-447.2 kg*m/s - 131 kg * v2_final) / 86 kg
Step 5: We can substitute this expression for v1_final into the momentum equation to solve for v2_final.
Step 6: Substitute the expression for v1_final in the momentum equation:
(86 kg * (-447.2 kg*m/s - 131 kg * v2_final) / 86 kg) + (131 kg * v2_final) = 0
Step 7: Simplify and solve for v2_final:
-447.2 kg*m/s - 131 kg * v2_final + 131 kg * v2_final = 0
-447.2 kg*m/s = 131 kg * v2_final
v2_final = -447.2 kg*m/s / 131 kg
Therefore, the final velocity of the fisherman is v1_final = (-447.2 kg*m/s - 131 kg * (-447.2 kg*m/s / 131 kg)) / 86 kg
And the final velocity of the rowboat is v2_final = -447.2 kg*m/s / 131 kg
Calculating the final velocities will give you the numerical values.