Is It True or False About Acceleration, Velocity, and Position?

[kchurchi]

Homework Statement

Conceptual acceleration, velocity, and position


Determine if the following statements are true or false. Remember that for a statement to be true it must be true in all cases.

a)If the velocity of an object is zero, then the acceleration of an object must also be zero.

b)If the initial velocity of an object is zero, then the initial speed of the object must be zero.

c)If the average velocity of an object is zero, then the average speed of the object must also be zero.

d)If the initial speed of an object is zero, then the initial position of the object must be the origin.

e)The acceleration of an object can be non-zero when the speed of the object is constant.

Homework Equations

none

The Attempt at a Solution

a)True - if the velocity of the object is zero, and acceleration is the derivative of velocity, then both velocity and acceleration are zero.

b)False - initial velocity has direction while initial speed has none. the initial velocity could be zero because its components are in opposing directions and are equal. However, if this were true, then speed would be non-zero because one of the components would no longer be negative.

c)False - the average velocity is the total displacement over total time. Average speed is the total distance traveled over total time. Average speed does not take into account the direction of the position vector, only its magnitude.

d)False - even though an object may be at rest to begin with, this does not necessarily place it at the origin. This varies from problem to problem.

e)False - if speed is constant then that means that velocity must be constant. This means that acceleration is zero since velocity is unchanging.

I don't understand where I am going wrong in my thought process. I have been going through this problem so many times it's ridiculous, though it should seem simple. Am I missing something obvious?

 

[physicsisgrea]

a) you are correct, a = dv/dt, v = 0 -> a = 0
b) no such things as "component of speed" because speed is a scalar quantity
c) you are correct
d) you are correct it is because the origin depends on our choice
e) "speed = constant" does not imply "velocity = constant". you can consider circular motion in uniform speed

 

[kchurchi]

Okay so I understand that
a) True (acceleration = 0)
b) You are right, speed has no components since it is a scalar. But using pythag. theorem to find the speed shows us that the components of velocity are squared, summed, and square rooted to find speed. Again, does my direction argument apply? I still think this is false.
c) False (correct)
d) False (correct)
e) This makes sense. Velocity can change direction (be non-constant) even though speed is constant.

 

[Doc Al]

a)True - if the velocity of the object is zero, and acceleration is the derivative of velocity, then both velocity and acceleration are zero.

What if you drop an object? What is its initial velocity? What is its acceleration?

b)False - initial velocity has direction while initial speed has none. the initial velocity could be zero because its components are in opposing directions and are equal. However, if this were true, then speed would be non-zero because one of the components would no longer be negative.

The speed is the magnitude of the velocity. If the velocity is zero that means that all components are zero.

c)False - the average velocity is the total displacement over total time. Average speed is the total distance traveled over total time. Average speed does not take into account the direction of the position vector, only its magnitude.

d)False - even though an object may be at rest to begin with, this does not necessarily place it at the origin. This varies from problem to problem.

Good.

e)False - if speed is constant then that means that velocity must be constant. This means that acceleration is zero since velocity is unchanging.

Careful: velocity is a vector and has direction. What if only the direction changes?

 

[kchurchi]

"What if you drop an object? What is its initial velocity? What is its acceleration?"

The initial velocity of the object is zero, but it's acceleration is the acceleration due to gravity, which is non-zero. This intuitively makes sense to me, but how does this fit the information that acceleration is the derivative of velocity? Does it make sense to say that even for an object at rest it is "accelerating" due to gravity (being pulled to the surface of the Earth by gravity)?.

"The speed is the magnitude of the velocity. If the velocity is zero that means that all components are zero."

Ohhhhhhh duh. I got it. In order for velocity to be zero all of its components are zero. For some reason I was trying to add them together

Thanks to both of you! I finally got the answer.

 

[CFede]

The initial velocity of the object is zero, but it's acceleration is the acceleration due to gravity, which is non-zero. This intuitively makes sense to me, but how does this fit the information that acceleration is the derivative of velocity? Does it make sense to say that even for an object at rest it is "accelerating" due to gravity (being pulled to the surface of the Earth by gravity)?.

This has no problem with the fact that acceleration is the derivative of the velocity. As it turns out, the velocity of a free falling body is v(t)=g*t, then:

a=dv/dt=g

which meas the acceleration is contant and equal to g. On the other hand, if you evaluate the velocity a t=0, then v(t=0)=g*0=0. So there is no problem there.

On the other hand, yes, it majes sense that an object at rest is accelerating, you just have to remember that, since it's accelerating it will remain at rest only for an instant (an infinitely small amount of time).

 

[Doc Al]

The initial velocity of the object is zero, but it's acceleration is the acceleration due to gravity, which is non-zero. This intuitively makes sense to me, but how does this fit the information that acceleration is the derivative of velocity?

The derivative is the rate of change. You cannot assume that the velocity is constant and thus always zero. (If that were true, then you'd be right: the acceleration would be zero if the velocity remained constant.)

Does it make sense to say that even for an object at rest it is "accelerating" due to gravity (being pulled to the surface of the Earth by gravity)?

Only if it's falling. An object just sitting on the floor isn't accelerating. It's being pulled down by gravity but also pushed up by the floor. No net force = no acceleration.