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. For what value of h does the collision occur at the instant when the first ball is at its highest point?

Verified step by step guidance

1. Determine the time it takes for the first ball to reach its highest point. At the highest point, the velocity of the first ball becomes zero. Use the kinematic equation: $v=v$₀-gt, where $v$ is the final velocity (0 at the highest point), $v$₀ is the initial velocity, $g$ is the acceleration due to gravity, and $t$ is the time. Solve for $t$: $t=\frac{v}{}$₀g.
2. Calculate the height of the first ball at its highest point using the kinematic equation: $y=v$₀t-12gt^2. Substitute $t=\frac{v}{}$₀g from Step 1 into this equation to find the maximum height $y$_max.
3. For the second ball, determine the time it takes to fall from height $h$ to the collision point. Use the kinematic equation: $y=h-\frac{1}{2}gt^2$. At the collision point, the height of the second ball equals the maximum height of the first ball, so set $y=y$_max.
4. Equate the time for the first ball to reach its highest point (from Step 1) to the time it takes for the second ball to fall to the collision point (from Step 3). Solve for $h$ in terms of $v$₀ and $g$.
5. Simplify the expression for $h$ to find the value of the initial height of the second ball such that the collision occurs at the highest point of the first ball. This will involve substituting $y$_max from Step 2 into the equation derived in Step 4.

Key Concepts

Kinematics

Kinematics is the branch of mechanics 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 scenario, understanding the kinematic equations is essential to determine the position of both balls over time, particularly the first ball's maximum height and the second ball's position as it falls.

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, with no other forces acting on it. In this case, the second rubber ball is dropped from rest, meaning it accelerates downward at a rate of approximately 9.81 m/s². This concept is crucial for calculating the height from which the second ball falls and determining when it will collide with the first ball.

Maximum Height

The maximum height of a projectile is the peak point reached during its upward motion, where its velocity becomes zero before it starts descending. For the first rubber ball shot upwards, this height can be calculated using the initial velocity and the acceleration due to gravity. Knowing this height is vital for finding the specific value of h at which the second ball will collide with the first ball at that instant.