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Let's suppose mass changes smoothly from 5 kg to 2 kg in 10 s. During this same interval of time the object goes from 0 m/s to 120 m/s (again let's say smoothly.) 1. Create mass and velocity as a function of time equations Can you use these to give the Mom(t) equation? 2. What is the momentum at time = 3 s? 3. What is the Force as a function of time equation? 4. What is the force at times 0, 3, 7 and 10 s?

Let's suppose mass changes smoothly from 5 kg to 2 kg in 10 s. During this same interval of time the object goes from 0 m/s to 120 m/s (again let's say smoothly.) 1. Create mass and velocity as a function of time equations Can you use these to give the Mom(t) equation? 2. What is the momentum at time = 3 s? 3. What is the Force as a function of time equation? 4. What is the force at times 0, 3, 7 and 10 s?

Glencoe Physics: Principles and Problems, Student Edition
Glencoe Physics: Principles and Problems, Student Edition
1st Edition
ISBN:9780078807213
Author:Paul W. Zitzewitz
Publisher:Paul W. Zitzewitz
Chapter9: Momentum And Its Conservation
Section9.1: Impulse And Momentum
Problem 2PP
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Momentum - A Calculus Perspective Suppose Momentum, (since m means mass we use Mom for momentum) is given as a function of time. This equation can be differentiated with respect to both time. Mom (t) = mv Up to this point mass has not changed as a function of time, but what if it did? dMom dv dt dt Mom' (t): = = V dm dt +m The student should recognize that acceleration is defined as dv/dt. Therefore, the second half of the differential is actually... Mom' (t) = ma = F(t) Which is, of course, Newton's 2nd law. So what does the first half of dmom/dt mean? Consider a rocket. Its engines have a certain thrust. This thrust is probably fairly constant but since the rocket is burning fuel it is losing mass over time. When the rocket is full, the thrust of the rocket will have to work against more inertia than when the rocket is almost empty. This means that the thrusters will actuall cause more acceleration as the rocket empties it's fuel tanks. This leads us back to Mom'(t) is defined as force. So let's try this in a problem.... Let's suppose mass changes smoothly from 5 kg to 2 kg in 10 s. During this same interval of time the object goes from 0 m/s to 120 m/s (again let's say smoothly.) 1. Create mass and velocity as a function of time equations Can you use these to give the Mom(t) equation? 2. What is the momentum at time = 3 s? 3. What is the Force as a function of time equation? 4. What is the force at times 0, 3, 7 and 10 s? Hint: It maybe helpful to start by determiong mass and velocity at 1 s intervals and use this to evaluate momentum..
Transcribed Image Text:Momentum - A Calculus Perspective Suppose Momentum, (since m means mass we use Mom for momentum) is given as a function of time. This equation can be differentiated with respect to both time. Mom (t) = mv Up to this point mass has not changed as a function of time, but what if it did? dMom dv dt dt Mom' (t): = = V dm dt +m The student should recognize that acceleration is defined as dv/dt. Therefore, the second half of the differential is actually... Mom' (t) = ma = F(t) Which is, of course, Newton's 2nd law. So what does the first half of dmom/dt mean? Consider a rocket. Its engines have a certain thrust. This thrust is probably fairly constant but since the rocket is burning fuel it is losing mass over time. When the rocket is full, the thrust of the rocket will have to work against more inertia than when the rocket is almost empty. This means that the thrusters will actuall cause more acceleration as the rocket empties it's fuel tanks. This leads us back to Mom'(t) is defined as force. So let's try this in a problem.... Let's suppose mass changes smoothly from 5 kg to 2 kg in 10 s. During this same interval of time the object goes from 0 m/s to 120 m/s (again let's say smoothly.) 1. Create mass and velocity as a function of time equations Can you use these to give the Mom(t) equation? 2. What is the momentum at time = 3 s? 3. What is the Force as a function of time equation? 4. What is the force at times 0, 3, 7 and 10 s? Hint: It maybe helpful to start by determiong mass and velocity at 1 s intervals and use this to evaluate momentum..
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