SUMMARY
The motion of a particle is defined by the equation x = t³ - 6t² - 36t - 40. The velocity is zero when the derivative of the position function, v(t) = 3t² - 12t - 36, equals zero, which occurs at t = 6 seconds. At this time, the acceleration, given by a(t) = 6t - 12, is calculated to be 24 ft/s². The total distance traveled when x = 0 is determined by solving the position equation for t, yielding a distance of 40 feet.
PREREQUISITES
- Understanding of calculus, specifically differentiation and integration.
- Familiarity with kinematic equations in physics.
- Knowledge of particle motion concepts, including velocity and acceleration.
- Ability to solve polynomial equations for real roots.
NEXT STEPS
- Study the application of derivatives in motion analysis.
- Learn how to solve polynomial equations using the Rational Root Theorem.
- Explore the relationship between velocity, acceleration, and displacement in physics.
- Investigate the implications of critical points on particle motion.
USEFUL FOR
Students studying calculus and physics, particularly those focusing on motion analysis and kinematics.
