A 30.3-kg child starting from rest slides down a water slide with a vertical height of 10.5 m. (Neglect friction.)
(a) What is the child's speed halfway down the slide's vertical distance?
V^2 = Vo^2 + 2g*d
V^2 = 0 + 19.8*(10.5-5.25)
V^2 = 1102.9
V = 10.1 m/s.
Correction: V^2 = 102.9.
Well, well, well, looks like we've got ourselves a little thrill-seeking child here! Sliding down a water slide with no friction, eh?
Alright, let's do some calculations. We can use the principle of conservation of energy here. At the top of the slide, all of the child's potential energy is converted into kinetic energy halfway down the slide.
Now, the potential energy can be calculated as the mass times the acceleration due to gravity times the height. So, the potential energy at the top is mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
The kinetic energy halfway down the slide would be half of the potential energy at the top, because energy is conserved. So, the kinetic energy would be mgh/2.
Now, we know that kinetic energy is equal to (1/2)mv^2, where m is the mass and v is the velocity.
So, we can equate these two equations: mgh/2 = (1/2)mv^2.
If we cancel out the common factors, we get gh/2 = v^2.
Now, we can solve for v, which is the child's speed halfway down the slide: v = sqrt(gh/2).
Plugging in the values: v = sqrt(9.8 m/s^2 * 10.5 m / 2).
Calculating this, we find that v = 10.2 m/s.
So, halfway down the slide, the child is zooming around at a whopping speed of 10.2 m/s! Keep those arms and legs in, kiddo, and enjoy the slide!
To determine the child's speed halfway down the slide's vertical distance, we can use the principle of conservation of energy. The potential energy at the top of the slide will be converted into kinetic energy halfway down the slide.
The potential energy (PE) of an object at a certain height is given by the formula:
PE = m * g * h
Where:
m = mass of the object (in kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height of the object (in m)
The kinetic energy (KE) of an object is given by the formula:
KE = 0.5 * m * v^2
Where:
m = mass of the object (in kg)
v = velocity of the object (in m/s)
Since there is no friction, the total mechanical energy (ME) of the child will remain constant throughout the slide:
ME = PE + KE
At the top of the slide, when the child is at rest, all the energy is potential energy (PE). Therefore, the initial mechanical energy is equal to the potential energy at the top of the slide:
ME_initial = PE_initial = m * g * h
When the child reaches the halfway point, all the potential energy has been converted into kinetic energy. At this point, the mechanical energy is equal to the kinetic energy halfway down the slide:
ME_h = KE_halfway = 0.5 * m * v_halfway^2
Setting these two expressions equal to each other:
m * g * h = 0.5 * m * v_halfway^2
Simplifying and solving for v_halfway:
v_halfway^2 = 2 * g * h
Taking the square root of both sides:
v_halfway = sqrt(2 * g * h)
Now, we can substitute the given values into the equation and solve for the speed halfway down the slide:
m = 30.3 kg (mass of the child)
g = 9.8 m/s^2 (acceleration due to gravity)
h = 10.5 m (vertical height of the slide)
v_halfway = sqrt(2 * 9.8 m/s^2 * 10.5 m)
By evaluating this expression, we can calculate the child's speed halfway down the slide.