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

Verified step by step guidance

1. Identify the initial velocity (v_0) of the boulder, which is given as 40.0 m/s.
2. Recognize that the acceleration due to gravity (g) is acting downwards and has a value of approximately 9.8 m/s^2.
3. Use the kinematic equation for vertical motion without air resistance, which is y = v_0t - 0.5gt^2, where y is the displacement from the initial position, v_0 is the initial velocity, g is the acceleration due to gravity, and t is the time.
4. Set the displacement y to zero to find the time when the boulder returns to its initial position. This gives the equation 0 = 40.0t - 0.5(9.8)t^2.
5. Solve the quadratic equation 0 = 40.0t - 4.9t^2 for t to find the times at which the displacement is zero. This will include the time at which the boulder is launched (t = 0) and the time it returns to the initial position after reaching its maximum height.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, kinematic equations can be used to analyze the boulder's motion, particularly to determine when its displacement returns to zero after being ejected.

Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is defined as the shortest distance from the initial to the final position, along with the direction. For the boulder, the displacement will be zero when it returns to its starting point after being ejected upward.

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object accelerates towards the Earth when in free fall, typically denoted as 'g' and approximately equal to 9.81 m/s². In this problem, the boulder's upward motion will decelerate due to gravity until it reaches its peak height, after which it will accelerate downward, ultimately affecting the time it takes to return to its initial position.