In anticipation of a long 7^∘ upgrade, a bus driver...

In anticipation of a long $7^{\circ}$ upgrade, a bus driver accelerates at a constant rate of $3 \mathrm{ft} / \mathrm{s}^{2}$ while still on a level section of the highway. Knowing that the speed of the bus is $60 \mathrm{mi} / \mathrm{h}$ as it begins to climb the grade and that the driver does not change the setting of his throttle or shift gears, determine the distance traveled by the bus up the grade when its speed has decreased to $50 \mathrm{mi} / \mathrm{h}$.

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Key Concepts

Uniformly Accelerated Motion

This concept involves the equations of motion under constant acceleration, such as v² = u² + 2ad, which relates the initial velocity (u), final velocity (v), acceleration (a), and displacement (d). It is fundamental to solving problems where an object is speeding up or slowing down at a constant rate, allowing one to determine any unknown variable when the others are known.

Force Decomposition on an Incline

When an object moves on an inclined plane, gravitational force must be resolved into components perpendicular and parallel to the plane. The parallel component (mg sin?) works against the motion when climbing an incline, affecting the net acceleration. This concept is crucial in understanding how the constant engine force results in a different net acceleration when transitioning from level ground to an upward slope.

Unit Conversion

Proper conversion of units is essential to ensure consistency in calculations. In dynamics problems, speeds given in miles per hour often need to be converted to feet per second when acceleration is specified in feet per second squared. This conversion enables the accurate application of kinematics formulas by aligning all units of measurement.

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