Two identical 1500 kg cars are moving perpendicular to each other. One moves with a speed of 25 m/s due north and the other moves at 15 m/s due east. What is the total momentum of the system?

Question

BlackTom AI · Accepted Answer

Momentum of each car is p = m v. For perpendicular velocities the total momentum is the vector sum; use components and the Pythagorean theorem.

1) Relevant concepts:
- Linear momentum p = m v for each object.
- For perpendicular vectors, resultant magnitude p_tot = sqrt(px^2 + py^2).
- Direction (angle) measured as θ = arctan(py/px); here “θ N of E” means angle northward from the east axis.

2) Step-by-step calculation:
- Mass of each car: m = 1500 kg.
- North-moving car speed: v_N = 25.0 m/s → p_N = m v_N = 1500 × 25 = 37,500 kg·m/s (north component).
- East-moving car speed: v_E = 15.0 m/s → p_E = m v_E = 1500 × 15 = 22,500 kg·m/s (east component).

3) Combine components (perpendicular):
- Magnitude: p_total = sqrt(p_N^2 + p_E^2)
  = sqrt((37,500)^2 + (22,500)^2)
  = sqrt(1,406,250,000 + 506,250,000)
  = sqrt(1,912,500,000)
  ≈ 43,732.1 kg·m/s ≈ 4.4 × 10^4 kg·m/s (rounded to two significant figures shown in choices).
- Direction (angle north of east): θ = arctan(p_N / p_E)
  = arctan(37,500 / 22,500)
  = arctan(1.6666667)
  ≈ 59.04° ≈ 59° N of E.

4) Why this answer is correct:
The calculation correctly treats momentum as vector quantities and uses perpendicular component addition. The magnitude 4.37×10^4 rounds to 4.4×10^4, and the arctan gives ~59°, matching “59° N of E.”

5) Why the other answer choices are incorrect:
- Any choice giving 6.0 × 10^4 kg·m/s uses scalar addition (37,500 + 22,500 = 60,000) which is wrong for vector sum.
- Choices listing 31° confuse angle measurement (31° would be east of north or the complement of 59°); it does not match arctan(p_N/p_E).

Conclusion: The correct total momentum is 4.4 × 10^4 kg·m/s at 59° N of E, which matches the provided answer.

Two identical 1500 kg cars are moving perpendicular to each other. One moves with a speed of 25 m/s due north and the other moves at 15 m/s due east. What is the total momentum of the system?单项选择题

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