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

Verified step by step guidance

1. Identify that the problem involves a boulder being projected vertically upward, which is a classic example of motion under gravity.
2. Recognize that the velocity of the boulder will be zero at the peak of its trajectory, where it momentarily stops before descending.
3. Use the kinematic equation for velocity: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mi>u</mi><mo>-</mo><mi>g</mi><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> is the final velocity (0 m/s at the peak), <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> is the initial velocity (40.0 m/s), <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is the acceleration due to gravity (9.8 m/s²), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time.
4. Set the final velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> to zero and solve the equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>40.0</mn><mo>-</mo><mn>9.8</mn><mi>t</mi></math>.
5. Rearrange the equation to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>40.0</mn><mn>9.8</mn></mfrac></math>. This will give you the time at which the velocity is zero.

Key Concepts

Kinematics in One Dimension

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one-dimensional motion, we analyze the position, velocity, and acceleration of an object along a straight line. For this problem, understanding how velocity changes over time under constant acceleration is crucial.

Free Fall and Gravity

Free fall refers to the motion of an object under the influence of gravitational force only. In this scenario, the boulder is subject to Earth's gravity, which causes a constant acceleration of approximately 9.81 m/s² downward. This acceleration affects the boulder's velocity as it moves upward and eventually comes to a stop.

Velocity and Acceleration Relationship

Velocity is the rate of change of an object's position, while acceleration is the rate of change of velocity. When an object is thrown upwards, its velocity decreases due to the negative acceleration (gravity) until it reaches zero at the peak of its trajectory. At this point, the object momentarily stops before descending.