Finding the Maximum Time for a Rocket's Flight Using Kinematics

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In summary: Then plug the solution t into T_{max} = T_{1} + t to get the final expression for T_max. In summary, the equation for T_max is T_max = T1 + t, where t is the solution to the equation g(\frac{1}{2}T1^2 + T1t - \frac{1}{2}t^2) = 0. This can be found by solving for t in the quadratic equation t^2 - 2T1t - T1^2 = 0 and plugging it into the equation for T_max.
  • #1
sherlockjones
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A rocket initially at rest accelerates with constant net acceleration B from t = 0 to t = T1 at which time the fuel is exhausted. Neglect air resistance. If the rocket's net acceleration, B, is equal to 1.0g, find an expression for the total time [itex] T_{max} [/itex] (from liftoff until it hits the ground).


So [tex] T_{max} = T_{1} + t [/tex]

[tex] \frac{1}{2}BT_{1}^{2} + BT_{1}t - \frac{1}{2}gt^{2} = 0 [/tex]

I know that [tex] t = \frac{BT_{1}}{g} [/tex]


What do I do from here? I got [tex] T_{max} = 2T_{1} = 2 t [/tex]


Thanks
 
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  • #2
Looks okay, since, when the rocket is left without any fuel, its motion is a free fall with y(t) = yo + vo t - 1/2 g t^2, where yo is the well-known height yo = y(T1) = 1/2 B T1^2 = 1/2 g T1^2, and v0 = BT1. You're on the right track. Now just solve for t, and plug it into Tmax = T1 + t.
 
  • #3
So is the equation [tex] \frac{1}{2}gt^{2} + gt^{2} - \frac{1}{2}gt^{2} [/tex]?
 
  • #4
The equation is [tex] \frac{1}{2}gT_{1}^{2} + gT_{1}t - \frac{1}{2}gt^{2} = 0 [/tex], as you already wrote. Now solve for t.
 
  • #5
If [tex] B = g [/tex] how do we get [tex] t^{2} - 2T_{1}t - T_{1}^{2} = 0 [/tex]?

I factored the equation: [tex] g(\frac{1}{2}T_{1}^{2} + T_{1}t - \frac{1}{2}t^{2}) = 0 [/tex]. I guess they used the relation that [tex] t = T_{1} [/tex] and multiplied both sides by 2?
 
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  • #6
sherlockjones said:
If [tex] B = g [/tex] how do we get [tex] t^{2} - 2T_{1}t - T_{1}^{2} = 0 [/tex]?

I factored the equation: [tex] g(\frac{1}{2}T_{1}^{2} + T_{1}t - \frac{1}{2}t^{2}) = 0 [/tex]. I guess they used the relation that [tex] t = T_{1} [/tex] and multiplied both sides by 2?

Again, solve the equation (i.e. find the roots of the parabola) [tex] \frac{1}{2}gT_{1}^{2} + gT_{1}t - \frac{1}{2}gt^{2} = 0 [/tex] for t. It is the only unknown.
 

Related to Finding the Maximum Time for a Rocket's Flight Using Kinematics

1. What is kinematics?

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion.

2. What are the three main components of kinematics?

The three main components of kinematics are position, velocity, and acceleration.

3. What is the difference between speed and velocity?

Speed is a scalar quantity that measures how fast an object is moving. Velocity, on the other hand, is a vector quantity that measures both the speed and direction of an object's motion.

4. How is acceleration related to velocity?

Acceleration is the rate of change of velocity. This means that when an object's velocity changes, it is experiencing acceleration.

5. How do you calculate displacement in kinematics?

Displacement is the change in an object's position. It can be calculated by subtracting the initial position from the final position.

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