Graphical Antiderivatives

[courtney1121]

A tow truck is pulling a car. Initially the force that the truck applies to the car is quite large. However, it decreases linearly to zero and remains at zero. Describe the how the acceleration, the velocity and the position change in time during this process.



Ok I have no idea where to start with this one. Any suggestions on where to start?

So I'm thinking that it's velocity and acceleration is increasing and it's position is increasing also...then the tow truck starts to decelerate causing velocity to decrease until the tow truck comes to rest...position is increasing the whole time. Actually, if it's at a constant velocity, there's no acceleration, then wouldn't that cause the force to decrease linearly to zero?

 

[americanforest]

Acceleration would drop along with Force until it was zero. Velocity would increase with the highest slope initially then its slope would drop until it became constant when F became zero. Position would begin going up with a slope equal to the initial velocity, then its slope would increase as velocity increased. When Velocity comes to a constant value, position would maintain the slope that is equivalent to the final velocity.

 

[courtney1121]

How do you get that though? And why is acceleration dropping? Is it because velocity is going to be constant?

 

[d_leet]

How do you get that though? And why is acceleration dropping? Is it because velocity is going to be constant?

How is force related to acceleration? Then how does acceleration realte to velocity and position?

 

[courtney1121]

Well if acceleration is 0 then there is no force acting on something or the forces might actually cancel out. Acceleration relates to velocity by the fact that when velocity is increasing, acceleration is increasing, and when velocity is decreasing, acceleration is decreasing in the opposite direction. When velocity is constant, there is no acceleration. Would position always increase? well unless you throw a ball up in the air and it comes back down.

 

[d_leet]

Well if acceleration is 0 then there is no force acting on something or the forces might actually cancel out.

I meant how does force relate to acceleration mathematically. Are they proportional?

Acceleration relates to velocity by the fact that when velocity is increasing, acceleration is increasing, and when velocity is decreasing, acceleration is decreasing in the opposite direction.

This isn't true. How do the two mathematically relate to each other? What is acceleration? What do we call a change in velocity over a change in time?

 

[courtney1121]

well F=ma so acceleration is proportional to force, right?

Mathematically acceleration relates to velocity by

a=v/t

so time will always be increasing...and if velocity is small then acceleration will be small and if velocity is increasing then acceleration will be large...

 

[d_leet]

well F=ma so acceleration is proportional to force, right?

Yes, this is correct.

Mathematically acceleration relates to velocity by

a=v/t

For average velocities this is true, but since we're dealing with calculus this isn't what you should be thinking.

so time will always be increasing...and if velocity is small then acceleration will be small and if velocity is increasing then acceleration will be large...

Large isn't the right word. Velocity is the antiderivative of acceleration, and acceleration is the derivative of velocity, so if velocity is increasing then the acceleration is positive.

 

[americanforest]

Think of it this way:

The value of Acceleration is the slope of the velocity/time graph. The value of the velocity time graph is the slope of the position graph. Thus, when Acceleration hits zero, velocity has zero slope and is thus constant. If Acceleration is positive, even if it is increasing, velocity is still increasing. The same applies for the relationship between velocity and position graphs related to time.