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Math  /  Algebra

Question51. (II) A high-speed 14 -car Italian train has a mass of 640 metric tons (640,000 kg)(640,000 \mathrm{~kg}). It can exert a maximum force of 400 kN horizontally against the tracks, whereas at maximum constant velocity (300 km/h)(300 \mathrm{~km} / \mathrm{h}), it exerts a force of about 150 kN . Calculate (a) its maximum acceleration, and (b) estimate the total force of friction and air resistance at top speed.

Studdy Solution

STEP 1

1. The mass of the train is 640,000 kg640,000 \, \text{kg}.
2. The maximum force exerted by the train is 400 kN400 \, \text{kN}.
3. The force exerted at maximum constant velocity is 150 kN150 \, \text{kN}.
4. The maximum constant velocity of the train is 300 km/h300 \, \text{km/h}.
5. We need to calculate the maximum acceleration and estimate the total force of friction and air resistance at top speed.

STEP 2

1. Calculate the maximum acceleration of the train.
2. Estimate the total force of friction and air resistance at top speed.

STEP 3

To find the maximum acceleration, use Newton's second law of motion, which states that the force is equal to the mass times the acceleration (F=ma F = ma ).
Given: - Maximum force F=400 kN=400,000 N F = 400 \, \text{kN} = 400,000 \, \text{N} - Mass m=640,000 kg m = 640,000 \, \text{kg}
The formula for acceleration a a is: a=Fm a = \frac{F}{m}
Substitute the given values: a=400,000 N640,000 kg a = \frac{400,000 \, \text{N}}{640,000 \, \text{kg}}

STEP 4

Calculate the acceleration: a=400,000640,000 m/s2 a = \frac{400,000}{640,000} \, \text{m/s}^2 a=0.625 m/s2 a = 0.625 \, \text{m/s}^2
The maximum acceleration of the train is 0.625 m/s2 \boxed{0.625 \, \text{m/s}^2} .

STEP 5

Estimate the total force of friction and air resistance at top speed.
At maximum constant velocity, the net force is zero because the train is not accelerating. Therefore, the force exerted by the train is equal to the total force of friction and air resistance.
Given: - Force exerted at maximum velocity Fmax velocity=150 kN=150,000 N F_{\text{max velocity}} = 150 \, \text{kN} = 150,000 \, \text{N}
The total force of friction and air resistance is 150,000 N \boxed{150,000 \, \text{N}} .

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