A 35.9 kg child starting from rest slides down a water slide with a vertical height of 20.3 m. (Neglect friction.)What is the child speed half way down
PE at top: mgh = 35.9*9.8*20.3 = 7142 J
She loses half that on the way down.
KE = 1/2 mv^2 = 3571
So, v^2 = 3571/35.9 = 99.47
Call it 100
So, v = 10
To determine the child's speed halfway down the water slide, we can make use of the principle of conservation of mechanical energy. At the top of the slide, the child has potential energy due to its height above the ground. As the child slides down, this potential energy is converted into kinetic energy.
The potential energy (PE) at the top of the slide can be calculated using the formula:
PE = m * g * h
where m is the mass of the child (35.9 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slide (20.3 m).
PE = 35.9 kg * 9.8 m/s^2 * 20.3 m = 7063.414 J
Since energy is conserved, this potential energy is equal to the kinetic energy (KE) halfway down the slide. Therefore:
KE = 7063.414 J
The kinetic energy of an object can be calculated using the formula:
KE = (1/2) * m * v^2
where v is the velocity of the child.
Now, we can rearrange the formula to solve for velocity:
v^2 = (2 * KE) / m
v^2 = (2 * 7063.414 J) / 35.9 kg
v^2 = 393.895 J/kg
Taking the square root of both sides:
v = sqrt(393.895 J/kg) ≈ 19.85 m/s
Therefore, the child's speed halfway down the water slide is approximately 19.85 m/s.
To determine the child's speed halfway down the water slide, we can use the principle of conservation of mechanical energy. At the top of the water slide, the child only has potential energy, which is given by the formula:
Potential Energy = mass x gravity x height
where mass is the mass of the child (35.9 kg), gravity is the acceleration due to gravity (9.8 m/s^2), and height is the vertical height of the water slide (20.3 m).
So, the potential energy at the top of the slide is:
Potential Energy = 35.9 kg x 9.8 m/s^2 x 20.3 m
Next, we can determine the child's potential energy halfway down the water slide. Since there is no friction involved (neglecting friction), the child's potential energy at the halfway point will be converted entirely into kinetic energy.
The formula for kinetic energy is given by:
Kinetic Energy = (1/2) x mass x velocity^2
where mass is the mass of the child (35.9 kg) and velocity is the speed of the child halfway down the slide.
Setting the potential energy at the top equal to the kinetic energy halfway down, we have:
Potential Energy = Kinetic Energy
35.9 kg x 9.8 m/s^2 x 20.3 m = (1/2) x 35.9 kg x (velocity halfway down)^2
Simplifying the equation and solving for velocity halfway down, we get:
(1/2) x 35.9 kg x (velocity halfway down)^2 = 35.9 kg x 9.8 m/s^2 x 20.3 m
(velocity halfway down)^2 = 2 x 9.8 m/s^2 x 20.3 m
Taking the square root of both sides to solve for velocity halfway down, we have:
velocity halfway down = √(2 x 9.8 m/s^2 x 20.3 m)
Now, calculating the right side of the equation:
velocity halfway down = √(2 x 9.8 m/s^2 x 20.3 m)
velocity halfway down ≈ √(398.44 m^2/s^2)
velocity halfway down ≈ 19.96 m/s
Therefore, the child's speed halfway down the water slide is approximately 19.96 m/s.