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

• sherlockjones
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.
sherlockjones
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 $T_{max}$ (from liftoff until it hits the ground).

So $$T_{max} = T_{1} + t$$

$$\frac{1}{2}BT_{1}^{2} + BT_{1}t - \frac{1}{2}gt^{2} = 0$$

I know that $$t = \frac{BT_{1}}{g}$$

What do I do from here? I got $$T_{max} = 2T_{1} = 2 t$$

Thanks

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.

So is the equation $$\frac{1}{2}gt^{2} + gt^{2} - \frac{1}{2}gt^{2}$$?

The equation is $$\frac{1}{2}gT_{1}^{2} + gT_{1}t - \frac{1}{2}gt^{2} = 0$$, as you already wrote. Now solve for t.

If $$B = g$$ how do we get $$t^{2} - 2T_{1}t - T_{1}^{2} = 0$$?

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

Last edited:
sherlockjones said:
If $$B = g$$ how do we get $$t^{2} - 2T_{1}t - T_{1}^{2} = 0$$?

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

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

## 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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