It takes a force of 53 kN on the lead car of a 16 -car passenger train with mass 9.1 ×10^5 kg to

step 1 In this question, you are given that there's a lead car that pulls a train. Okay, so a lead car,

It takes a force of 53 kN on the lead car of a 16 -car...

It takes a force of 53 $\mathrm{kN}$ on the lead car of a 16 -car passenger train with mass $9.1 \times 10^{5} \mathrm{kg}$ to pull it at a constant 45 $\mathrm{m} / \mathrm{s}$ $(101 \mathrm{mi} / \mathrm{h})$ on level tracks. (a) What power must the locomotive provide to the lead car? (b) How much more power to the lead car than calculated in part (a) would be needed to give the train an acceleration of 1.5 $\mathrm{m} / \mathrm{s}^{2}$ , at the instant that the train has a speed of 45 $\mathrm{m} / \mathrm{s}$ on level tracks? (c) How much more power to the lead car than that calculated in part (a) would be needed to move the train up a 1.5$\%$ grade (slope angle $\alpha=\arctan 0.015 )$ at a constant 45 $\mathrm{m} / \mathrm{s} ?$

It takes a force of 53 $\mathrm{kN}$ on the lead car of a 16 -car passenger train with mass $9.1 \times 10^{5} \mathrm{kg}$ to pull it at a constant 45 $\mathrm{m} / \mathrm{s}$ $(101 \mathrm{mi} / \mathrm{h})$ on level tracks. (a) What power must the locomotive provide to the lead car? (b) How much more power to the lead car than calculated in part (a) would be needed to give the train an
acceleration of 1.5 $\mathrm{m} / \mathrm{s}^{2}$ , at the instant that the train has a speed of
45 $\mathrm{m} / \mathrm{s}$ on level tracks? (c) How much more power to the lead car than that calculated in part (a) would be needed to move the train up a 1.5$\%$ grade (slope angle $\alpha=\arctan 0.015 )$ at a constant 45 $\mathrm{m} / \mathrm{s} ?$

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Key Concepts

Inclined Plane Dynamics

Motion on an inclined plane involves additional considerations due to the gravitational component acting along the slope. The effective force against gravity is determined by the weight of the object and the sine of the incline angle, which necessitates extra power to overcome this gravitational pull when moving at a steady speed.

Additional Power for Acceleration

When an object accelerates, additional force is necessary beyond that required to maintain constant velocity. This extra force, calculated as mass times acceleration, requires an extra amount of power at a given speed, emphasizing how dynamic conditions change the energy demands of a system.

Mechanical Power

Mechanical power is the rate at which work is done or energy is transferred in a system. It is calculated as the product of force and velocity, linking how quickly a force performs work on a moving object.

Newton's Second Law

Newton's Second Law explains the relationship between force, mass, and acceleration, stating that the net force acting on an object equals its mass multiplied by its acceleration. This principle is crucial for determining the additional force required when an object accelerates.

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