A large boulder is ejected vertically upward from a volcano with an initial speed of $40.0$ m/s. Ignore air resistance. At what time is it moving at $20.0$ m/s downward?

Verified step by step guidance

1. Identify the initial conditions: The boulder is ejected with an initial speed of 40.0 m/s upward. We need to find the time when its speed is 20.0 m/s downward.
2. Understand the motion: The boulder is subject to gravitational acceleration, which is approximately 9.8 m/s² downward. This will slow it down as it moves upward, stop it momentarily, and then accelerate it downward.
3. Use the kinematic equation for velocity: The velocity of an object under constant acceleration can be described by the equation: $v=v$_i-gt, where $v$ is the final velocity, $v$_i is the initial velocity, $g$ is the acceleration due to gravity, and $t$ is the time.
4. Set up the equation: Substitute the known values into the equation. The final velocity $v$ is -20.0 m/s (downward), $v$_i is 40.0 m/s, and $g$ is 9.8 m/s². The equation becomes: $-20.0=40.0-9.8t$.
5. Solve for time $t$: Rearrange the equation to solve for $t$. This involves isolating $t$ on one side of the equation: $t=\frac{(40.0--20.0)}{9.8}$. Calculate the value of $t$.

Key Concepts

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They are essential for calculating the position, velocity, and time of an object in motion. In this problem, the equations help determine the time at which the boulder reaches a specific velocity while moving downward.

Gravitational Acceleration

Gravitational acceleration is the constant acceleration experienced by objects due to Earth's gravity, approximately 9.81 m/s² downward. This concept is crucial for understanding the motion of the boulder as it moves upward and then downward, affecting its velocity and time of travel.

Velocity Direction

Velocity direction indicates whether an object is moving upward or downward. In this problem, understanding that the boulder initially moves upward and then downward is key to determining when its velocity is 20.0 m/s downward, requiring consideration of both magnitude and direction.