A rocket rises vertically, from rest, with an acceleration of 3.2 m/s² until it runs out of fuel at an altitude of 725 m. After this point, its acceleration is that of gravity, downward. What maximum altitude does the rocket reach?

Verified step by step guidance

1. Determine the velocity of the rocket at the moment it runs out of fuel. Use the kinematic equation: $v^2\; =\; v\_0^2\; +\; 2a\; d$, where $v\_0\; =\; 0$ (initial velocity), $a\; =\; 3.2\; \backslash (\backslash text\{\; m/s\}\backslash )^2$ (acceleration), and $d\; =\; 725\; \backslash (\backslash text\{\; m\}\backslash )$ (distance traveled while fuel is burning). Solve for $v$.
2. Once the rocket runs out of fuel, it continues to move upward due to its velocity at that point. Its motion is now under the influence of gravity alone, with an acceleration of $a\; =\; -9.8\; \backslash (\backslash text\{\; m/s\}\backslash )^2$. Use the kinematic equation $v^2\; =\; v\_0^2\; +\; 2a\; d$ again, where $v\; =\; 0$ (velocity at the maximum altitude), $v\_0$ is the velocity calculated in step 1, and $a\; =\; -9.8\; \backslash (\backslash text\{\; m/s\}\backslash )^2$. Solve for $d$, which represents the additional height the rocket travels after running out of fuel.
3. Add the altitude at which the rocket runs out of fuel ($725\; \backslash (\backslash text\{\; m\}\backslash )$) to the additional height calculated in step 2 to find the maximum altitude reached by the rocket.
4. Verify the units and ensure all calculations are consistent with the SI system. Double-check the signs of acceleration and velocity to ensure they align with the direction of motion.
5. Summarize the process: The maximum altitude is the sum of the altitude at fuel exhaustion and the additional height gained while decelerating under gravity. This ensures a clear understanding of the problem-solving approach.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for understanding how the rocket accelerates upward due to the thrust generated by its engines until it runs out of fuel.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this scenario, we can use these equations to calculate the rocket's velocity at the moment it runs out of fuel and subsequently determine its maximum altitude by considering the motion under the influence of gravity alone.

Free Fall and Gravity

Once the rocket runs out of fuel, it enters free fall, where the only force acting on it is gravity, which accelerates it downward at approximately 9.81 m/s². Understanding free fall is essential for calculating how high the rocket ascends after it exhausts its fuel and begins to decelerate until it reaches its peak altitude.