What is the Equation for Resistive Force and Speed in a Straight-Line Race?

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SUMMARY

The equation for resistive force acting on an object moving in a straight line is defined as F = -k*m*V^2, where k is a constant and m is the mass of the object. In the context of a speed skater, the final speed after crossing the finish line can be expressed as Vf = Vi / (1 + Vi*k*t), where Vi is the initial speed and t is the time elapsed. The mass cancels out in the derivation, confirming that the resistive force is proportional to the square of the speed. This relationship is crucial for understanding motion under resistive forces.

PREREQUISITES
  • Understanding of Newton's second law of motion
  • Familiarity with basic calculus concepts
  • Knowledge of resistive forces and their mathematical representation
  • Concept of exponential decay in motion
NEXT STEPS
  • Study the derivation of the equation for resistive forces in physics
  • Explore the implications of resistive forces in real-world applications, such as vehicle dynamics
  • Learn about differential equations and their role in modeling motion
  • Investigate the effects of varying the constant k on speed over time
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Physics students, engineers, and anyone interested in the dynamics of motion under resistive forces will benefit from this discussion.

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Consider an object which the net force is a resistive force proportional to the square of its speed. For example: assume that the resistive force acting on a speed skater is F=-k*m*V^2, where k is a constant and m is the skater's mass. The skater crosses the finish line of a straight-line race with speed V(i) and the slows down by coasting on his skates. Show that his speed at time "t", any time after the finish line is equal to Vf=Vi/(1+Vi*k*t).

I know that the masses cancel out and I get this m dv/dt= kmV^2 since "mass times acceleration equals force".

But now what? How does the Vi gets introduced?

Thank you in advance!
 
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