A rubber ball is shot straight up from the ground with speed v₀. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. At what height above the ground do the balls collide? Your answer will be an algebraic expression in terms of h, v₀, and g.

Verified step by step guidance

1. Define the motion equations for both balls. For the first ball (moving upward with initial velocity v₀), its height as a function of time t is given by: $y$₁=v0t-12gt2. For the second ball (dropped from height h), its height as a function of time t is: $y$₂=h-12gt2.
2. Set the heights of the two balls equal to each other to find the time t at which they collide: $v_{0}t-\frac{1}{2}g{t}^2=h-\frac{1}{2}g{t}^2$.
3. Simplify the equation by canceling out the common term $\frac{1}{2}g{t}^2$ on both sides. This leaves: $v_{0}t=h$.
4. Solve for the time t at which the collision occurs: $t=\frac{h}{v_0}$.
5. Substitute the value of t back into the height equation for either ball to find the height of collision. Using the first ball's equation: $y=v_0\frac{h}{v_0}-\frac{1}{2}g{\frac{h}{v_0}}^2$. Simplify this expression to get the final algebraic expression for the height of collision.

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, and acceleration. In this problem, kinematic equations will be used to determine the positions of both balls as functions of time, which is essential for finding the collision height.

Acceleration due to Gravity (g)

Acceleration due to gravity, denoted as g, is the acceleration experienced by an object due to the gravitational force exerted by the Earth. Near the Earth's surface, g is approximately 9.81 m/s² and acts downward. This constant will affect the motion of both balls, influencing their velocities and positions over time.

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from another moving object. In this scenario, the motion of the two balls must be analyzed relative to each other to determine the point at which they collide. Understanding how their velocities and positions change over time is crucial for solving the problem.