A locomotive with a power capability of 1.5 MW can accelerate a train from a speed of 10 m / s

step 1 Well, average power of the locomotive, p average, is equal to 1 .5 megawatts. 1 .5 megawatt. An

A locomotive with a power capability of 1.5 MW can...

A locomotive with a power capability of $1.5 \mathrm{MW}$ can accelerate a train from a speed of $10 \mathrm{~m} / \mathrm{s}$ to $25 \mathrm{~m} / \mathrm{s}$ in $6.0 \mathrm{~min}$. (a) Calculate the mass of the train. Find (b) the speed of the train and (c) the force accelerating the train as functions of time (in seconds) during the $6.0$ min interval. (d) Find the distance moved by the train during the interval.

Key Concepts

Mechanical Power

Mechanical power is the rate at which work is done or energy is transferred. In dynamics problems like this one, power is expressed as the product of force and instantaneous velocity, meaning that when power is constant, the force applied must adjust as the velocity changes. Understanding mechanical power is crucial because it governs how the applied force will vary during the acceleration of the train.

Newton's Second Law

Newton's Second Law states that the net force acting on an object equals the product of its mass and acceleration (F = ma). This law forms the basis for relating the force exerted by the locomotive to the resulting acceleration of the train. It is fundamental in solving problems where the mass of the system must be determined from given acceleration conditions.

Variable Acceleration and Differential Equations

In situations where the applied force or power is not constant, acceleration becomes a variable that depends on time or velocity. This requires setting up differential equations that relate the rate of change of velocity (acceleration) to the changing force. Solving these differential equations through integration is essential to obtain functions describing the velocity and force as functions of time.

Integration in Kinematics

When acceleration is not constant, integration is used to derive the velocity function from the acceleration function and subsequently the displacement function from the velocity function. This process is a key technique in kinematics, enabling the calculation of distance traveled under variable acceleration conditions, as seen in systems with constant power output.

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