A Frog Collides with a Hanging Rock: Calculating Maximum Height

Scenario:

A 1.1-kg frog, moving to the right at 10.7 m/s but not looking where it is going, plows into a 9.7-kg rock hanging motionless from a long string. The frog bounces off the hanging rock, recoiling to the left at 8.64 m/s. Calculate the maximum height h to which the rock rises as it swings to the right.

Given Data:

Mass of frog (m1) = 1.1 kg

Initial velocity of frog (v1_initial) = 10.7 m/s to the right

Mass of hanging rock (m2) = 9.7 kg

Initial velocity of rock (v2_initial) = 0 m/s (hanging motionless)

Final velocity of frog (v1_final) = 8.64 m/s to the left

Calculations:

Initially, the momentum of the frog is equal to the total momentum of the system (frog + rock) after the collision. We can use conservation of momentum to find the final velocity of the rock.

m1 * v1_initial = m1 * v1_final + m2 * v2_final

Final answer:

To find the maximum height to which the rock rises after being struck by a frog, one must first apply conservation of momentum to find the rock's velocity just after the collision, and then use energy conservation to convert the kinetic energy of the rock to gravitational potential energy at the maximum height.

Explanation:

To calculate the maximum height h to which the rock rises after being struck by the frog, we must first use the law of conservation of momentum for the collision. The initial momentum of the system (frog plus rock) is solely due to the moving frog. After the collision, the frog and rock will have momenta that add up to this initial momentum. The conservation of momentum can be described by the equation:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

Where m1 and m2 are the masses of the frog and rock respectively, and v1_initial and v2_initial are the initial velocities of the frog and rock (with the rock's initial velocity being 0 as it is initially at rest). The final velocities are v1_final and v2_final for the frog and rock respectively. Using the given masses and velocities (frog's initial and final velocities), we can solve for the rock's final velocity just after the collision.

After finding the rock's final velocity, we then use energy conservation to determine the height h. We know that initially, the rock has kinetic energy due to its velocity post-collision, and at the maximum height, all this kinetic energy would be converted into potential energy (as the velocity will be 0 at the peak of its ascent). This conversion can be described by the equation:

KE_initial = PE_final

1/2 * m2 * v2_final^2 = m2 * g * h

Rearranging this for height h, we find:

h = v2_final^2 / (2 * g

We now have all the information needed to compute h, once v2_final is determined from the momentum equation. Remember that g is the acceleration due to gravity, which is approximately 9.81 m/s^2