What is the momentum of the pellet?

[emily081715]

Homework Statement

A burst of compressed air pushes a pellet out of a blowpipe. The force exerted by the air on the pellet is given by F(t)=F0e(−t/τ), where τ is called a time constant because it has units of time.
What is the momentum of the pellet after an interval equal to one time constant has elapsed?
Express your answer in terms of the variables F0, τ, and exponential constant e.

Homework Equations

The Attempt at a Solution

i honestly am completely lost on this question, i assumed it would just be F0e(−t) but that's not correct. help?

 

[I like Serena]

H

Homework Statement

A burst of compressed air pushes a pellet out of a blowpipe. The force exerted by the air on the pellet is given by F(t)=F0e(−t/τ), where τ is called a time constant because it has units of time.
What is the momentum of the pellet after an interval equal to one time constant has elapsed?
Express your answer in terms of the variables F0, τ, and exponential constant e.

Homework Equations

The Attempt at a Solution

i honestly am completely lost on this question, i assumed it would just be F0e(−t) but that's not correct. help?

Hey Emily!

We have momentum p=mv.
Furrhermore we have force F=ma.
And speed v is the integral of acceleration a with respect to time.
Suppose we integrate the expression. What will we get?

 

[emily081715]

HHey Emily!

We have momentum p=mv.
Furrhermore we have force F=ma.
And speed v is the integral of acceleration a with respect to time.
Suppose we integrate the expression. What will we get?

I'm completely lost how to integrate it though,F(t)=F0e(−t/τ) is not like a function I'm use to working with

 

[I like Serena]

I'm completely lost how to integrate it though,F(t)=F0e(−t/τ) is not like a function I'm use to working with

The exponential function is an odd one. Its integral looks very similar to its derivative.
What would the derivative be?

Oh, and for the record:
$$\int Fdt=\int madt=mv=p$$

 

[emily081715]

The exponential function is an odd one. Its integral looks very similar to its derivative.
What would the derivative be?

wouldn't the derivative be e(−t/τ) +F0e(−t/τ) ?

 

[I like Serena]

wouldn't the derivative be e(−t/τ) +F0e(−t/τ) ?

How so?

Note that the derivative of $e^x$ is $e^x$. That leaves applying the chain rule. How familiar are you with the chain rule?

 

[emily081715]

How so?

Note that the derivative of $e^x$ is $e^x$. That leaves applying the chain rule. How familiar are you with the chain rule?

i was assuming there was also product rule too since fo is a constant

 

[I like Serena]

i was assuming there was also product rule too since fo is a constant

Nope. No product rule for constants.
Or rather, the derivative of a constant is zero, causing the corresponding term to be zero as well.

 

[I like Serena]

The derivative is:
$$F'(t) = F_0 \cdot -\frac 1\tau e^{-t/\tau}$$
This helps to find the integral, which is:
$$p(\tau) = \int_0^\tau F\,dt = F_0 \cdot -\tau e^{-t/\tau}\Big|_0^\tau$$

 

[emily081715]

i solved the whole question already