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Solve the differential equation to find velocity v as a function of time t if v = 0 when t = 0. The differential equation models the motion of two people on a toboggan after consideration of the forces of gravity, friction, and air resistance. 12.5 = 43.6 - 1.25v dt

Solve the differential equation to find velocity v as a function of time t if v = 0 when t = 0. The differential equation models the motion of two people on a toboggan after consideration of the forces of gravity, friction, and air resistance. 12.5 = 43.6 - 1.25v dt

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
**Solve the Differential Equation for Velocity in a Toboggan System** To find the velocity \( v \) as a function of time \( t \), solve the following differential equation given that \( v = 0 \) when \( t = 0 \). This equation reflects the motion of two people on a toboggan, considering the forces of gravity, friction, and air resistance: \[ 12.5 \frac{dv}{dt} = 43.6 - 1.25v \] **Equation Breakdown:** - **Left Side:** \( 12.5 \frac{dv}{dt} \) represents the rate of change of velocity, multiplied by a constant (12.5), which might relate to the mass or other systemic factors affecting acceleration. - **Right Side:** \( 43.6 - 1.25v \) combines a constant force (possibly due to gravity direction) and a damping force proportional to velocity \( v \) (1.25v), which could represent friction or air resistance opposing motion. The goal is to solve this first-order linear differential equation to express velocity \( v \) as a function of time \( t \). This mathematical model helps in understanding the dynamics of the toboggan's motion.
Transcribed Image Text:**Solve the Differential Equation for Velocity in a Toboggan System** To find the velocity \( v \) as a function of time \( t \), solve the following differential equation given that \( v = 0 \) when \( t = 0 \). This equation reflects the motion of two people on a toboggan, considering the forces of gravity, friction, and air resistance: \[ 12.5 \frac{dv}{dt} = 43.6 - 1.25v \] **Equation Breakdown:** - **Left Side:** \( 12.5 \frac{dv}{dt} \) represents the rate of change of velocity, multiplied by a constant (12.5), which might relate to the mass or other systemic factors affecting acceleration. - **Right Side:** \( 43.6 - 1.25v \) combines a constant force (possibly due to gravity direction) and a damping force proportional to velocity \( v \) (1.25v), which could represent friction or air resistance opposing motion. The goal is to solve this first-order linear differential equation to express velocity \( v \) as a function of time \( t \). This mathematical model helps in understanding the dynamics of the toboggan's motion.
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