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 after being ejected is the boulder moving at $20.0$ m/s upward?

Verified step by step guidance

1. Identify the known values: initial velocity \( v_0 = 40.0 \text{ m/s} \), final velocity \( v = 20.0 \text{ m/s} \), and acceleration due to gravity \( a = -9.8 \text{ m/s}^2 \) (negative because it acts downward).
2. Use the kinematic equation that relates initial velocity, final velocity, acceleration, and time: \( v = v_0 + at \).
3. Substitute the known values into the equation: \( 20.0 = 40.0 + (-9.8)t \).
4. Rearrange the equation to solve for time \( t \): \( t = \frac{20.0 - 40.0}{-9.8} \).
5. Calculate the value of \( t \) using the rearranged equation to find the time at which the boulder is moving at 20.0 m/s upward.

Key Concepts

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. For vertical motion, these equations relate initial velocity, final velocity, acceleration due to gravity, time, and displacement. In this problem, the equation v = u + at is used, where v is final velocity, u is initial velocity, a is acceleration (gravity), and t is time.

Acceleration due to Gravity

Gravity is a constant force that accelerates objects towards the Earth at approximately 9.81 m/s². In this scenario, the boulder is subject to this acceleration, which affects its velocity over time. Understanding gravity's role is crucial for calculating the time it takes for the boulder to reach a specific velocity.

Initial and Final Velocity

Initial velocity is the speed at which an object starts its motion, while final velocity is the speed at a later time. In this problem, the boulder starts with an initial velocity of 40.0 m/s and we need to find the time when its velocity decreases to 20.0 m/s due to gravity's deceleration effect.