How Does Rain Affect the Momentum and Speed of a Moving Cart?

In summary: If the initial mass is m0 and mass is being gained at rate ##\lambda##, what is the mass at time t?If the initial mass is m0 and mass is being gained at rate ##\lambda##, the mass at time t would be m1 = m0 + \lambda*t.
  • #1
brinstar
62
1

Homework Statement


An empty cart travels at 15 m/s as the rain begins to fall. The rain goes into the cart, and adds 10 kg of mass per second.

1) Will the rain cause the momentum of the cart to change?
2) Write an equation for the speed of the cart as a function of time.
3) How fast will this cart be moving after 200 seconds?

Homework Equations


dp/dt = 0
delta p = 0 = m(dv/dt) + v(dm/dt)
a = dv/dt

The Attempt at a Solution


1) No, a change in momentum in this system = 0 (as given by my teacher during the quiz). So mass will change due to rain, but velocity will adjust and there will be negative acceleration that will slow down the cart. Momentum, overall, will not change and will stay the same.

2) v(t) = 15 m/s - 0.15t m/s^2

0 = m(dv/dt) + v(dm/dt)
-150 mkg/s^2 = 1,000 kg (dv/dt)
-0.15 m/s^2 = dv/dt = a
at = v

3) v(200) = 15 m/s - 0.15 m/s^2 (200s) = 15 m/s - 30 m/s = -15 m/s
*If the cart is not on a slope or hill, it is probably already at rest by 200 s. IT has slowed considerably due to the rain.

This was a quiz my teacher gave, and I only got the first question right. I now have a 33% grade (unfortunately, it's all or nothing => no partial credit...) :/ She also refuses to post a solution sheet or give us the answers, and insists that we find the solutions ourselves to enhance our understanding of the material... Can someone please guide me through what exactly I did wrong on this 3-part quiz? Thanks for any help!
 
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  • #2
Where does the acceleration of -.15m/s2 come from?
To answer this question, you need to know the mass of the cart.
 
  • #3
haruspex said:
Where does the acceleration of -.15m/s2 come from?
To answer this question, you need to know the mass of the cart.
Oh sorry! The cart is 1000 kg, I forgot to include that. My bad.
 
  • #4
brinstar said:
Oh sorry! The cart is 1000 kg, I forgot to include that. My bad.
Ok, so use the result in part 1. What is the initial momentum? What is the mass at time t?
 
  • #5
haruspex said:
Ok, so use the result in part 1. What is the initial momentum? What is the mass at time t?
The initial momentum is m(vi) = 1000 kg (15 m/s) = 15,000 kgm/s, and I'm assuming that stays the same because momentum change = 0.

As for the mass at time t, there's 10 kg per second, so 200 seconds later, there's 2,000 more g. So the mass at 200s is 3,000 kg, I woud suppose.
 
  • #6
brinstar said:
The initial momentum is m(vi) = 1000 kg (15 m/s) = 15,000 kgm/s, and I'm assuming that stays the same because momentum change = 0.
Yes.
brinstar said:
As for the mass at time t, there's 10 kg per second, so 200 seconds later, there's 2,000 more g. So the mass at 200s is 3,000 kg, I woud suppose.
Yes, but (2) requires this expressed as a function of t, not at a specific time.
When you have that, use it and the conservation of momentum to find the velocity at time t.
 
  • #7
haruspex said:
Yes.

Yes, but (2) requires this expressed as a function of t, not at a specific time.
When you have that, use it and the conservation of momentum to find the velocity at time t.

So wouldn't it be v(t) = 15 - .15t m/s?
 
  • #8
brinstar said:
So wouldn't it be v(t) = 15 - .15t m/s?
Why do you think it would be that? I asked you earlier where you got the 0.15 from.

(Ask yourself what this would make the velocity after 200 seconds.)
 
  • #9
haruspex said:
Why do you think it would be that? I asked you earlier where you got the 0.15 from.

(Ask yourself what this would make the velocity after 200 seconds.)

I got -0.15 from looking for dv/dt in the thrust equation. It's also acceleration, so multiplying that by time gives me velocity. So the original speed (15 m/s) minus the potential change in speed (-0.15t) would be what I assume gets the number. Multiplied out, however, it gets to -15 m/s, which is it's either at rest or going backwards.

um... v(t) = ... I don't know how to bring mass into that. Should I use a momentum equation?
 
  • #10
brinstar said:
I got -0.15 from looking for dv/dt in the thrust equation.
But you used the initial mass and initial velocity. Both change over time, so the acceleration will change.
brinstar said:
um... v(t) = ... I don't know how to bring mass into that. Should I use a momentum equation?
If the initial mass is m0 and mass is being gained at rate ##\lambda##, what is the mass at time t?
 
  • #11
haruspex said:
But you used the initial mass and initial velocity. Both change over time, so the acceleration will change.

If the initial mass is m0 and mass is being gained at rate ##\lambda##, what is the mass at time t?

ah okay

m(t) = 1,000 + 10t kg

^would that be it?
 
  • #12
brinstar said:
ah okay

m(t) = 1,000 + 10t kg

^would that be it?
Yes. What equation does that allow you to write using conservation of momentum?
 
  • #13
haruspex said:
Yes. What equation does that allow you to write using conservation of momentum?

p(t) = (1,000 + 10t kg) (15 - .15t m/s)?
 
  • #14
brinstar said:
p(t) = (1,000 + 10t kg) (15 - .15t m/s)?
No, the acceleration is not a constant -.15m/s2. Please abandon that.
You know the initial momentum, you know momentum is conserved, you know the mass at time t. So what is the velocity at time t?
 
  • #15
If we think about conservation of momentum, then the rain water also do have momentum. In actual world the cart will be lowered with increased load, thus changed in direction.

Just like in inelastic collision.
 
  • #16
azizlwl said:
If we think about conservation of momentum, then the rain water also do have momentum. In actual world the cart will be lowered with increased load, thus changed in direction.
For the purposes of the question, only horizontal momentum is interesting. The rain arrives, we assume, vertically.
 
  • #17
haruspex said:
For the purposes of the question, only horizontal momentum is interesting. The rain arrives, we assume, vertically.

Hmmm... Initial momentum is 15,000 kgm/s. m(t) = 1,000 + 10t kg.

v(t) = 15,000 kgm/s / (1000 + 10t kg)

Is it this one?
 
  • #18
brinstar said:
Hmmm... Initial momentum is 15,000 kgm/s. m(t) = 1,000 + 10t kg.

v(t) = 15,000 kgm/s / (1000 + 10t kg)

Is it this one?
Bingo.
 
  • #19
haruspex said:
Bingo.

BOO-YAH!

And as follows:

v(200) = 15,000 kgm/s / (1000 + 10(200) kg) = 5,000 m/s?
 
  • #20
brinstar said:
BOO-YAH!

And as follows:

v(200) = 15,000 kgm/s / (1000 + 10(200) kg) = 5,000 m/s?
Right idea, wrong arithmetic.
 
  • #21
haruspex said:
Right idea, wrong arithmetic.

Ah, my bad lol 5 m/s. I don't know why I left in the zeroes.
 

FAQ: How Does Rain Affect the Momentum and Speed of a Moving Cart?

What is cart thrust and momentum?

Cart thrust and momentum refers to the force and motion of a cart, typically in a linear direction. It is a measure of how much force is required to move the cart, and how fast it will move in response to that force.

How is cart thrust and momentum related to Newton's Laws of Motion?

Cart thrust and momentum are directly related to Newton's Laws of Motion. According to Newton's Second Law, the force applied to an object is equal to its mass multiplied by its acceleration, or F=ma. In the case of a cart, the force is the thrust and the acceleration is the change in momentum. Additionally, Newton's Third Law states that for every action, there is an equal and opposite reaction. Therefore, the cart exerts a force on the ground, and the ground exerts an equal and opposite force on the cart, allowing it to move.

How is cart thrust and momentum calculated?

Cart thrust and momentum can be calculated using the formula p=mv, where p is momentum, m is mass, and v is velocity. To calculate the thrust, we can use the formula F=ma where F is force, m is mass, and a is acceleration. By combining these two equations, we can calculate the thrust required to achieve a certain momentum.

How does cart thrust and momentum affect the motion of the cart?

Cart thrust and momentum directly affect the motion of the cart. The greater the thrust, the faster the cart will accelerate and the greater the change in momentum will be. Similarly, the greater the mass of the cart, the more force is required to achieve a certain change in momentum. Additionally, friction and other external forces can also affect the motion of the cart and must be taken into consideration.

How can cart thrust and momentum be applied in real-world scenarios?

Cart thrust and momentum have many practical applications in our daily lives. For example, understanding these concepts is crucial in designing vehicles, calculating the force required to move heavy objects, and even in sports such as track and field. They are also important in understanding the concept of inertia and how objects resist changes in motion. Additionally, studying cart thrust and momentum can help us understand and improve the efficiency of machines and systems that utilize these principles.

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