Velocity Decay Curve for a Rocket

In summary, the conversation is about solving a problem involving the integral of a decay curve and finding the initial velocity of a rocket in order for it to travel a certain distance in a given time interval. The formula D=Ar(1-e-t/r) is used, with D, r, and t being known values and A being the unknown. The individual is stuck on how to calculate A but has been unable to find the answer after hours of searching. They suggest replacing a known value with a calculated one to solve for A.
  • #1
pete321
3
0
Homework Statement
suggest a new initial velocity A of the rocket
Relevant Equations
$$\large \int_0^{t_L} A e^{\frac{-t}{\tau}} \text{d}t = A \tau \left( 1- e^{\frac{-t}{\tau}} \right)$$
I am trying to solve the problem below. I have previously calculated from 0 to 4 seconds how far the rocket will travel in each second. I am stuck now as to how to start this problem. I have searched but unable to find the answer. Do i need to rearrange this? A is currently 14 which does not get the rocket to travel 15m in 4 seconds. Once i know how to start this i want to solve on my own so i understand how to complete this.

The integral of the decay curve of the form:
$$Ae^{\frac{-t}{\tau}}$$ This can be expressed as follows:

$$\large \int_0^{t_L} A e^{\frac{-t}{\tau}} \text{d}t = A \tau \left( 1- e^{\frac{-t}{\tau}} \right)$$

$$a = 14$$

$$\tau = 1.6$$

$$ t = 4$$Given this information, suggest a new initial velocity A of the rocket, which will allow the rocket to travel 15m in the same time interval of 0 to t=4.
 
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  • #2
Looks to me like you have an equation with 3 knowns (t, r, and distance) and 1 unknown (a). Can you solve for the 1 unknown?
 
  • #3
Thank you for your reply. No this is where i am confused. Is A now 15. As when t is 4 seconds the distance will be 15 not 14? I just don't understand how to calculate this. I have been searching for hours with no luck
 
  • #4
replacing 14 with 10.22 in the formula =15.0097. So would A be 10.22? If so just need to work out how to calculate it
 
  • #5
pete321 said:
replacing 14 with 10.22 in the formula =15.0097. So would A be 10.22? If so just need to work out how to calculate it
Like I said, you have 3 knowns and 1 unknown.
D=Ar(1-e-t/r), and we know D, r, and t.
So you rearrange the equation to solve for A.
 
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FAQ: Velocity Decay Curve for a Rocket

What is a Velocity Decay Curve for a Rocket?

A Velocity Decay Curve for a Rocket is a graphical representation of how the velocity of a rocket changes over time as it travels through the atmosphere. It shows the rate at which the rocket's velocity decreases due to air resistance.

How is the Velocity Decay Curve calculated?

The Velocity Decay Curve is calculated by plotting the velocity of the rocket on the y-axis and time on the x-axis. The data is collected through experiments or simulations and then plotted to create the curve.

What factors affect the Velocity Decay Curve for a Rocket?

The main factors that affect the Velocity Decay Curve for a Rocket are the rocket's initial velocity, the density of the atmosphere, and the rocket's shape and size. Other factors such as wind speed and temperature can also have an impact.

Why is the Velocity Decay Curve important for rocket design?

The Velocity Decay Curve is important for rocket design because it helps engineers understand how the rocket will perform in different atmospheric conditions. It also allows them to optimize the rocket's shape and size to minimize air resistance and maximize velocity.

How does the Velocity Decay Curve change as the rocket travels through the atmosphere?

The Velocity Decay Curve typically shows a steep decline in velocity at the beginning as the rocket encounters the densest part of the atmosphere. As it travels higher, the curve will level off as the air density decreases. Eventually, the curve will plateau as the rocket reaches a constant velocity due to the balance between air resistance and thrust.

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