Nicole throws a ball straight up. Chad watches the ball from a window 5.0 m above the point where Nicole released it. The ball passes Chad on the way up, and it has a speed of 10 m/s as it passes him on the way back down. How fast did Nicole throw the ball?

Verified step by step guidance

1. Step 1: Identify the known values and variables. Nicole throws the ball straight up, and Chad observes it from a window 5.0 m above the release point. The ball has a speed of 10 m/s as it passes Chad on the way down. We need to determine the initial velocity of the ball (Nicole's throw). The acceleration due to gravity is constant at \(-9.8\, \text{m/s}^2\).
2. Step 2: Use the kinematic equation \(v^2 = v_0^2 + 2a \Delta y\) to relate the velocity, initial velocity, acceleration, and displacement. Here, \(v\) is the velocity of the ball as it passes Chad on the way down (10 m/s), \(v_0\) is the initial velocity we are solving for, \(a\) is the acceleration due to gravity \(-9.8\, \text{m/s}^2\), and \(\Delta y\) is the displacement (5.0 m).
3. Step 3: Rearrange the kinematic equation to solve for \(v_0\): \(v_0^2 = v^2 - 2a \Delta y\). Substitute the known values: \(v = 10\, \text{m/s}\), \(a = -9.8\, \text{m/s}^2\), and \(\Delta y = 5.0\, \text{m}\).
4. Step 4: Take the square root of \(v_0^2\) to find \(v_0\): \(v_0 = \sqrt{v^2 - 2a \Delta y}\). Ensure the signs are correctly handled, as the acceleration is negative (gravity acts downward).
5. Step 5: Interpret the result. The calculated \(v_0\) represents the speed at which Nicole threw the ball upward. This value accounts for the ball's motion under the influence of gravity and its observed speed at Chad's window.

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, acceleration, and time. In this scenario, understanding the kinematic equations is essential to relate the initial velocity of the ball, its final velocity as it passes Chad, and the distance it travels.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the ball's motion, gravitational potential energy at its highest point is converted to kinetic energy as it falls back down. This concept helps in determining the initial speed of the ball by equating the potential energy at the peak with the kinetic energy when it passes Chad.

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. This constant affects the ball's upward and downward motion, influencing its velocity at different points. Understanding how gravity impacts the ball's trajectory is crucial for calculating the initial throw speed.