A massless rope hangs vertically from a tree. A child runs toward the
rope at speed 5m/s and grabs the end of the rope. This causes the child
to lift off of the ground, swing up into the air, and then to come back
down. The total time that the child is off the ground is 5sec. Treat the
child as a point mass of zero size and all angles as small.
(a) What is the length of the rope?
(b) What is the child’s maximum height off the ground during their swing?
To solve this problem, we can use the concepts of conservation of energy and projectile motion.
(a) To find the length of the rope, we need to know the time it takes for the child to reach maximum height while swinging on the rope. Since the total time the child is off the ground is 5 seconds and assuming the child reaches maximum height symmetrically, we can divide the total time in half: 5 seconds / 2 = 2.5 seconds.
Next, let's consider the energy of the system. At the lowest point, just before the child grabs the rope, the child has only kinetic energy, given by K = (1/2)mv^2, where m is the mass of the child (assumed to be negligible as mentioned in the problem) and v is the child's speed. Therefore, the energy at this point is K_initial = (1/2)(0)v^2 = 0.
At the highest point, the child has only potential energy, given by P = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height. Therefore, the energy at the highest point is P_max = mgh_max.
Since energy is conserved, the sum of the initial kinetic energy and the maximum potential energy must be equal: K_initial + P_max = 0 + mgh_max. Simplifying this equation, we get h_max = v^2 / (2g).
Since we know the child's speed (v = 5 m/s) and the acceleration due to gravity (g = 9.8 m/s^2), we can calculate the child's maximum height off the ground (h_max) using the formula mentioned above.
(b) To find the child's maximum height off the ground, substitute the given values into the formula:
h_max = (5 m/s)^2 / (2 * 9.8 m/s^2)
= 25 m^2/s^2 / 19.6 m/s^2
≈ 1.28 m
Therefore, the child's maximum height off the ground during the swing is approximately 1.28 meters.
Note: In this problem, we have neglected the mass of the child and assumed all angles are small. Additionally, we assume no energy losses due to friction or air resistance, which may slightly affect the accuracy of the solution.