A Boeing 787-9 jetliner has a mass of 245,000 kg including passengers. Its two engines produce

step 1 In the first part of this problem, we are going to calculate the engine's power of the jet liner

A Boeing 787-9 jetliner has a mass of 245,000 kg including...

A Boeing 787-9 jetliner has a mass of $245,000 \mathrm{~kg}$ including passengers. Its two engines produce a combined thrust force of $642 \mathrm{kN},$ and the aircraft cruises at $913 \mathrm{~km} / \mathrm{h}$ in level flight $-\mathrm{in}$ which case drag from the air is the only force the plane needs to overcome. Find the engines' power output (a) while cruising and (b) when it's climbing at a $23.0^{\circ}$ angle at $622 \mathrm{~km} / \mathrm{h}$. Assume air resistance doesn't change-although in reality it's greater at the higher speed.

Key Concepts

Mechanical Power

Mechanical power is the rate at which work is performed, calculated as the product of force and the velocity in the direction of the force. In problems like this, the engine’s power output is determined by multiplying the effective force (such as thrust or drag) by the speed of the aircraft.

Force-Velocity Relationship

The relation between force and velocity is fundamental in mechanics; when an object moves at constant speed, the engine’s force output exactly counters other forces like drag. The power required in such a case is simply the force (which equals the drag in steady level flight) multiplied by the velocity.

Vector Decomposition on an Incline

When motion occurs on an incline, forces need to be resolved into components along the direction of motion. For an aircraft climbing at an angle, the engine must overcome not only the drag force but also the component of gravitational force acting against the climb, which is determined by multiplying the weight of the aircraft by the sine of the climb angle.

Climbing Dynamics and Gravitational Work

Climbing requires additional work to increase the aircraft’s gravitational potential energy. This additional power requirement is calculated based on the vertical component of the velocity and the gravitational force acting on the aircraft, highlighting how energy is used not only to counter drag but also to lift the aircraft against gravity.

Unit Conversions in Mechanics

Accurate unit conversion is essential in physics problems to ensure consistency. For example, converting speeds from kilometers per hour to meters per second before calculating power helps maintain proper SI units, ensuring the resulting power value is correct.

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