Physics of Iceboat Motion: Comparing Two Masses Under Equal Force
Question:
Two iceboats (one of mass m, one of mass 2m) hold a race on a frictionless, horizontal frozen lake. Both iceboats start at rest, and the wind exerts the same constant force on both. Compare their:
- Final velocity
- Kinetic energy
- Momentum
- Impulse
Which boat wins the race and why?
Answer:
Step 1: Time to Reach the Finish Line
Using the kinematic equation:
d = (1/2)·a·t²
Acceleration (a) from Newton’s second law:
a = F / m
For Boat 1 (mass m):
a₁ = F/m ⇒ t₁ = √(2md / F)
For Boat 2 (mass 2m):
a₂ = F/2m ⇒ t₂ = √(4md / F) = √2·t₁
The lighter boat (mass m) reaches the finish line faster.
Step 2: Final Velocity
Velocity after time t:
v = a·t
Boat 1 (mass m):
v₁ = (F/m)·√(2md / F) = √(2Fd / m)
Boat 2 (mass 2m):
v₂ = (F/2m)·√(4md / F) = √(Fd / 2m)
Boat 1 (mass m) has a higher final velocity than Boat 2.
Step 3: Kinetic Energy
Formula:
KE = (1/2)·m·v²
Boat 1:
KE₁ = (1/2)·m·(2Fd / m) = Fd
Boat 2:
KE₂ = (1/2)·2m·(Fd / 2m) = Fd
Both iceboats have the same amount of kinetic energy at the finish line: Fd.
Step 4: Momentum
Formula:
p = m·v
Boat 1:
p₁ = m·√(2Fd / m) = √(2mFd)
Boat 2:
p₂ = 2m·√(Fd / 2m) = 2·√(mFd / 2) = √(4mFd / 2) = √(2mFd)
The heavier boat (mass 2m) has greater momentum.
Step 5: Impulse
Impulse (J) is given by:
J = F·t
Boat 1:
J₁ = F·√(2md / F) = √(2mFd)
Boat 2:
J₂ = F·√(4md / F) = √(4mFd) = 2·√(mFd)
Boat 2 (mass 2m) experiences a larger impulse.
Final Conclusions:
- ✅ Both boats have equal kinetic energy: Fd
- ❌ Boat 1 does NOT have more kinetic energy
- ❌ Boat 1 does NOT have more momentum
- ❌ Boat 2 has lower final velocity
- ✅ Boat 2 experiences larger impulse
Correct Statements:
✔ Both iceboats have the same amount of kinetic energy.
✔ The iceboat with mass 2m experiences a larger impulse.
✔ Both iceboats have the same amount of kinetic energy.
✔ The iceboat with mass 2m experiences a larger impulse.