To find the vertical acceleration of the water-skier, you can use the formula:
F = ma
where F is the net force acting on the water-skier, m is the mass of the water-skier, and a is the acceleration.
In this case, you know the mass of the water-skier and the force being applied by the rope, so you can solve for the acceleration.
First, you need to find the net force acting on the water-skier. The net force is equal to the sum of all forces acting on the water-skier. In this case, there are two forces acting on the water-skier: the force of gravity and the force being applied by the rope.
The force of gravity is equal to the weight of the water-skier, which is equal to the mass of the water-skier multiplied by the acceleration due to gravity (g):
F_gravity = m * g = 68 kg * 9.8 m/s^2 = 666.4 N
The net force acting on the water-skier is equal to the sum of the forces of gravity and the rope force:
F_net = F_gravity + F_rope = 666.4 N + 870 N = 1536.4 N
Then, you can use the formula for acceleration to solve for the acceleration of the water-skier:
a = F_net / m = 1536.4 N / 68 kg = 22.6 m/s^2
So the vertical acceleration of the water-skier is approximately 22.6 m/s^2.
It's important to note that this calculation assumes that there is no air resistance acting on the water-skier. If there is air resistance present, the actual acceleration of the water-skier may be different.