Tarzan of mass 80kg swings from rest on a rope of length 10m that is horizontal when he starts at the bottom of his swing his speed is 14m.second he picks up jane sitting on the ground in an inelastic collision jane has a mass of 50kg they then swing upwards as one unit...calculate the combined speed of Tarzan and Jane just after he picks her up.
.plz can anyone tell me which year is this question paper plz asap
November 2009
To solve this problem, we can use the principle of conservation of mechanical energy.
1. Before Tarzan picks up Jane, he is initially at the bottom of his swing with a speed of 14 m/s. The initial potential energy (PE) is zero since he is at the bottom. The initial kinetic energy (KE) can be calculated using the formula KE = 0.5 * mass * velocity^2:
KE_tarzan_initial = 0.5 * 80 kg * (14 m/s)^2
2. When Tarzan picks up Jane, their masses combine, but their total energy remains the same. Thus, the total energy before and after the collision is equal.
3. After the collision, both Tarzan and Jane swing upwards together. At the highest point of their swing, their speed will be at a minimum (zero).
4. At the highest point, all the initial kinetic energy will be converted into potential energy.
Let's calculate the maximum height (h) that Tarzan and Jane reach:
KE_tarzan_initial = PE_max
0.5 * 80 kg * (14 m/s)^2 = 80 kg * 9.8 m/s^2 * h
Simplifying the equation:
0.5 * 80 kg * (14 m/s)^2 = 80 kg * 9.8 m/s^2 * h
0.5 * 80 * 196 = 80 * 9.8 * h
7840 = 784 * h
h = 10 m
Therefore, the maximum height Tarzan and Jane reach is 10 meters above the bottom.
5. Now we can calculate the combined speed of Tarzan and Jane just after he picks her up. At the highest point, their speed will be zero, so we only need to consider their potential energy at that point.
PE_max = KE_tarzan_and_jane_final
0.5 * (80 kg + 50 kg) * v_final^2 = (80 kg + 50 kg) * 9.8 m/s^2 * 10 m
0.5 * 130 kg * v_final^2 = 1260 kg * m^2/s^2
Simplifying the equation:
v_final^2 = (1260 kg * m^2/s^2) / 130 kg
v_final^2 = 9.692307692307693 m^2/s^2
v_final ≈ 3.11 m/s
Therefore, the combined speed of Tarzan and Jane just after he picks her up is approximately 3.11 m/s.
To solve this problem, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system (Tarzan, Jane, and the rope) is conserved throughout their swing.
Initially, when Tarzan is at the bottom of his swing, he has a speed of 14 m/s. At this point, all of his mechanical energy is in the form of kinetic energy.
Next, Tarzan picks up Jane, and they swing upwards as a single unit. Since this is an inelastic collision, no energy is lost during the process. Thus, the total mechanical energy before and after the collision remains the same.
The total mechanical energy of the system is given by the sum of the kinetic energy and gravitational potential energy. At the bottom of the swing, all of the energy is in the form of kinetic energy:
Initial mechanical energy = 1/2 * mass * velocity^2 = 1/2 * 80 kg * (14 m/s)^2 = 7840 J
As Tarzan swings upwards, he gains gravitational potential energy, which is given by m * g * h, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the distance above the starting point. Since the swing is horizontal, Tarzan's height at any point is the length of the rope minus his distance from the starting point:
h = total rope length - distance traveled by Tarzan = 10 m - 5 m = 5 m
Gravitational potential energy = mass * g * h = 80 kg * 9.8 m/s^2 * 5 m = 3920 J
Since the total mechanical energy is conserved, we can equate the initial mechanical energy to the sum of the final kinetic energy and gravitational potential energy:
Initial mechanical energy = Final mechanical energy
7840 J = Final kinetic energy + Gravitational potential energy
Since Jane is now part of the system, the combined mass of Tarzan and Jane is 80 kg + 50 kg = 130 kg.
Final mechanical energy = 1/2 * total mass * (combined speed)^2
So, the equation becomes:
7840 J = 1/2 * 130 kg * (combined speed)^2 + 3920 J
Rearranging the equation, we can solve for the combined speed:
1/2 * 130 kg * (combined speed)^2 = 3920 J
Multiplying both sides by 2:
130 kg * (combined speed)^2 = 7840 J - 3920 J
130 kg * (combined speed)^2 = 3920 J
(combined speed)^2 = 3920 J / 130 kg
(combined speed)^2 = 30.154
(combined speed) = sqrt(30.154)
(combined speed) ≈ 5.49 m/s
Therefore, the combined speed of Tarzan and Jane just after he picks her up is approximately 5.49 m/s.