A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

Verified step by step guidance

1. Define the motion of the rocket and the bolt. The rocket is moving upward with constant acceleration, and the bolt detaches after 4 seconds. The bolt then follows a free-fall motion under gravity after detachment. Use kinematic equations to analyze both motions.
2. Determine the velocity of the rocket (and the bolt) at the moment the bolt detaches. Use the kinematic equation: $v=u+at$, where $u$ is the initial velocity (0 m/s), $a$ is the rocket's acceleration, and $t$ is 4 seconds.
3. Calculate the height of the bolt at the moment it detaches from the rocket. Use the kinematic equation: $s=ut+\frac{1}{2}at^2$, where $s$ is the height, $u$ is 0 m/s, $a$ is the rocket's acceleration, and $t$ is 4 seconds.
4. Analyze the motion of the bolt after detachment. The bolt starts with an initial upward velocity equal to the rocket's velocity at detachment and is subject to gravitational acceleration ($g=9.8 m/s^2$). Use the kinematic equation: $s=vt+\frac{1}{2}at^2$ to find the total displacement of the bolt during its free fall.
5. Combine the equations for the rocket's motion and the bolt's motion to solve for the rocket's acceleration. Use the fact that the total time for the bolt to hit the ground after detachment is 6 seconds, and the displacement of the bolt relative to the ground is zero when it lands.

Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In the context of the rocket, this means that the rocket's speed increases uniformly as it ascends. This concept is crucial for determining the rocket's motion and the time it takes for the bolt to fall after detaching.

Free Fall

Free fall describes the motion of an object under the influence of gravity alone, with no other forces acting on it. Once the bolt detaches from the rocket, it enters free fall, accelerating downward at approximately 9.81 m/s². Understanding free fall is essential for calculating the time it takes for the bolt to hit the ground after it falls off the rocket.

Kinematic Equations

Kinematic equations are mathematical formulas that describe the motion of objects under constant acceleration. These equations relate displacement, initial velocity, final velocity, acceleration, and time. They are vital for solving problems involving the rocket's ascent and the bolt's descent, allowing us to find the rocket's acceleration based on the given time intervals.