A physics student on Planet Exidor throws a ball, and it follows the parabolic trajectory shown in FIGURE EX4.13. The ball's position is shown at 1 s intervals until t = 3s. At t = 1s, the ball's velocity is v = (2.0 i + 2.0 j) m/s. What is the value of g on Planet Exidor?

Verified step by step guidance

1. Identify the known values: At t = 1 s, the velocity of the ball is \( \mathbf{v} = 2.0 \mathbf{i} + 2.0 \mathbf{j} \) m/s. The ball's motion is parabolic, indicating that it is under the influence of a constant gravitational acceleration \( \mathbf{g} \). The horizontal component of velocity remains constant, while the vertical component changes due to gravity.
2. Understand the relationship between acceleration and the change in velocity: The vertical acceleration \( g \) is responsible for the change in the vertical component of velocity \( v_y \). The equation for vertical velocity is \( v_y = v_{y0} - g t \), where \( v_{y0} \) is the initial vertical velocity and \( t \) is the time.
3. Determine the vertical velocity at a later time: From the problem, the ball's position is shown at 1-second intervals. At \( t = 2 \) s, the vertical velocity \( v_y \) can be calculated using the change in position or inferred from the trajectory data provided in the figure (not shown here).
4. Use the change in vertical velocity to calculate \( g \): The change in vertical velocity \( \Delta v_y \) over a time interval \( \Delta t \) is related to \( g \) by the equation \( g = \frac{\Delta v_y}{\Delta t} \). Substitute the known values of \( \Delta v_y \) and \( \Delta t \) (e.g., \( \Delta t = 1 \) s) to find \( g \).
5. Conclude the calculation: Once \( g \) is determined, it represents the gravitational acceleration on Planet Exidor. Ensure that the units are consistent (e.g., m/s²) and verify the result using the trajectory data provided in the figure.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to the force of gravity. It follows a curved path known as a parabola, characterized by horizontal and vertical components of motion. Understanding this concept is crucial for analyzing the trajectory of the ball on Planet Exidor, as it helps in determining how the initial velocity and gravitational acceleration affect the ball's path.

Velocity and Acceleration

Velocity is a vector quantity that describes the rate of change of an object's position, including both speed and direction. Acceleration, particularly gravitational acceleration (g), is the rate of change of velocity over time. In this scenario, knowing the initial velocity of the ball allows us to calculate the gravitational acceleration on Planet Exidor by analyzing how the velocity changes as the ball moves along its trajectory.

Gravitational Acceleration

Gravitational acceleration (g) is the acceleration experienced by an object due to the force of gravity acting on it. On Earth, this value is approximately 9.81 m/s², but it can vary on different planets. To find the value of g on Planet Exidor, one can use the kinematic equations of motion, which relate the initial velocity, time, and displacement of the ball to determine how gravity influences its trajectory.