A large boulder is ejected vertically upward from a volcano with an initial speed of $40.0$ m/s. Ignore air resistance. When is the displacement of the boulder from its initial position zero?

Verified step by step guidance

1. Identify that the problem involves projectile motion, specifically vertical motion under gravity. The initial speed is given as 40.0 m/s, and we need to find when the displacement is zero.
2. Understand that displacement is zero when the boulder returns to its initial position after being ejected upwards. This involves calculating the time taken for the boulder to reach its highest point and then return back down.
3. Use the kinematic equation for vertical motion: $v=u-gt$, where $v$ is the final velocity (0 m/s at the highest point), $u$ is the initial velocity (40.0 m/s), $g$ is the acceleration due to gravity (approximately 9.8 m/s²), and $t$ is the time to reach the highest point.
4. Solve for $t$ using the equation: $t=\frac{u}{g}$. This gives the time to reach the highest point. Since the motion is symmetrical, the total time for the boulder to return to its initial position is twice this value.
5. Calculate the total time for the boulder to return to its initial position by multiplying the time to reach the highest point by 2. This will give you the time when the displacement is zero.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. In this scenario, the boulder follows a vertical path, and its motion can be analyzed using kinematic equations to determine when it returns to its initial position.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. For vertical motion, these equations can be used to calculate displacement, velocity, and time. The equation s = ut + 0.5at^2 helps determine when the displacement is zero, where s is displacement, u is initial velocity, a is acceleration, and t is time.

Acceleration due to Gravity

Acceleration due to gravity is a constant force acting on objects near Earth's surface, typically approximated as 9.8 m/s² downward. This force affects the boulder's upward motion, slowing it down until it stops and reverses direction, eventually returning to its starting point, which is crucial for calculating the time of zero displacement.