Find the maximum speed of the elevato

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SUMMARY

The maximum speed of the elevator is calculated to be \( \frac{1}{4} \alpha t_0 \), where \( \alpha \) represents the acceleration coefficient and \( t_0 \) is the time duration of the elevator's motion. To find the total distance traveled by the elevator, the velocity function must be derived from the acceleration function \( a_y(t) = \alpha - 2\alpha \frac{t}{t_0} \). The correct integration of the velocity function over the interval from 0 to \( t_0 \) is necessary to obtain the total distance, which is not \( \frac{1}{4} \alpha t_0^2 \) as initially attempted.

PREREQUISITES
  • Understanding of kinematics, specifically motion under constant acceleration
  • Familiarity with calculus, particularly integration techniques
  • Knowledge of the relationship between acceleration, velocity, and displacement
  • Ability to manipulate algebraic expressions involving variables like \( \alpha \) and \( t_0 \)
NEXT STEPS
  • Learn how to derive the velocity function from an acceleration function
  • Study the principles of definite integrals in calculus for calculating displacement
  • Explore the concepts of motion in elevators and the forces acting on objects in non-inertial frames
  • Practice problems involving kinematic equations and integration to solidify understanding
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Students studying physics, particularly those focusing on mechanics and kinematics, as well as educators looking for examples of motion analysis in elevators.

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Homework Statement



A person of mass mp stands on a scale in an elevator of mass me . The scale reads the magnitude of the force F exerted on it from above in a downward direction. Starting at rest at t=0 the elevator moves upward, coming to rest again at time t=t_0. The downward acceleration of gravity is g . The acceleration of the elevator during this period is shown graphically above and is given analytically by ay(t)=α−2αt_0t

a) Find the maximum speed of the elevator. Express your answer in terms of α and t0

b) Find the total distance traveled by the elevator. Express your answer in terms of α and t0 (enter alpha for α and t_0 for t0)

Homework Equations





The Attempt at a Solution



I got part a right (answer is 1/4*alpha*t_0) but I don't know how to get b I tried integrating the velocity with repsect to t with the limits of t_0 and 0 and I got 1/4*a*t_0^2, but that is not right. Can someone please tell me what I;m doing wrong?
 
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What is the velocity function and how do you integrate it?
 

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