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. Determine the ball's velocity at t = 0 s, 2s, and 3s.

Verified step by step guidance

1. Step 1: Understand the problem. The ball follows a parabolic trajectory, which indicates that it is under the influence of gravity. The velocity at any point can be determined by analyzing the horizontal and vertical components of motion separately. The horizontal velocity remains constant, while the vertical velocity changes due to gravitational acceleration.
2. Step 2: Break the given velocity at t = 1s into components. The velocity is given as \( \mathbf{v} = (2.0 \mathbf{i} + 2.0 \mathbf{j}) \, \text{m/s} \). Here, \( 2.0 \mathbf{i} \) is the horizontal component \( v_x \), and \( 2.0 \mathbf{j} \) is the vertical component \( v_y \). Note that gravity only affects the vertical component.
3. Step 3: Use the vertical motion equation to determine the vertical velocity at different times. The equation for vertical velocity is \( v_y = v_{y0} - g t \), where \( v_{y0} \) is the initial vertical velocity, \( g \) is the acceleration due to gravity on Planet Exidor, and \( t \) is the time. At t = 1s, \( v_y = 2.0 \text{ m/s} \). Rearrange the equation to find \( g \): \( g = \frac{v_{y0} - v_y}{t} \).
4. Step 4: Calculate the velocity at t = 0s, 2s, and 3s. For t = 0s, the velocity is \( \mathbf{v} = v_x \mathbf{i} + v_{y0} \mathbf{j} \). For t = 2s and t = 3s, use the horizontal velocity \( v_x = 2.0 \text{ m/s} \) (constant) and the vertical velocity equation \( v_y = v_{y0} - g t \) to find the vertical components at those times.
5. Step 5: Combine the horizontal and vertical components to express the velocity as a vector at each time. For example, at t = 2s, \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \), where \( v_x = 2.0 \text{ m/s} \) and \( v_y \) is calculated using the vertical motion equation. Repeat this process for t = 3s.

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. The horizontal motion is uniform, while the vertical motion is influenced by gravitational acceleration, typically approximated as 9.81 m/s² downward.

Velocity Vector

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time, incorporating both speed and direction. In this context, the velocity vector is expressed in terms of its components, such as v = (vx i + vy j) m/s, where 'i' and 'j' represent the horizontal and vertical directions, respectively. Understanding how to decompose and analyze these components is crucial for determining the ball's velocity at different times.

Kinematic Equations

Kinematic equations are mathematical formulas that relate an object's displacement, initial velocity, final velocity, acceleration, and time. These equations are essential for solving problems involving motion, particularly when acceleration is constant, as in projectile motion. By applying these equations, one can calculate the velocity of the ball at various time intervals, given its initial conditions and the effects of gravity.