During a test run on an airport runway, a new race car...

During a test run on an airport runway, a new race car reaches a speed of 258.4 mph from a standing start. The car accelerates with constant acceleration and reaches this speed mark at a distance of $612.5 \mathrm{~m}$ from where it started. What was its speed after one-fourth, one-half, and three-fourths of this distance?

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Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's rate of change of velocity remains the same throughout its motion. This condition allows the use of straightforward kinematic formulas to relate displacement, velocity, and time, making it easier to analyze and predict the behavior of the moving object.

Kinematic Equations

Kinematic equations are the mathematical formulas used to describe the motion of objects, especially under constant acceleration. One core equation is v² = u² + 2as, where v represents the final velocity, u the initial velocity, a the constant acceleration, and s the displacement. This equation is particularly useful when an object starts from rest, allowing for the relationship between speed and distance to be clearly defined.

Relationship Between Velocity and Distance

When an object accelerates from a standstill under constant acceleration, its final velocity is proportional to the square root of the distance traveled. This is derived from the equation v² = 2as, which shows that if the object travels a fraction of the total distance, its speed is the final speed multiplied by the square root of that fraction. This concept is crucial for determining the speed at various intervals of the total distance.

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