At an amusement park, a swimmer uses a water slide to enter the main pool.
If the swimmer starts at rest, slides without friction, and descends through a vertical height of 2.41 m, what is her speed at the bottom of the slide?
Well, if the swimmer is using a water slide, they're definitely going to be making a splashy entrance into the main pool! But let's solve this sliding mystery, shall we?
To find the swimmer's speed at the bottom of the slide, we can use a little concept called conservation of energy. As the swimmer glides down the slide, her potential energy at the top gets converted into kinetic energy at the bottom.
The equation for conservation of energy is:
Potential Energy (PE) + Kinetic Energy (KE) = Total Energy
Since the swimmer is starting at rest, her initial kinetic energy is zero:
PE + 0 = PE (because KE = 0)
So, all we need to do is equate the potential energy at the top to the final kinetic energy at the bottom:
mgh = 0.5mv^2
Where:
m is the mass of the swimmer (which we don't know)
g is the acceleration due to gravity (roughly 9.8 m/s^2)
h is the vertical height of the slide (2.41 m)
v is the final velocity we're looking for
We can cancel out the mass (m) on both sides of the equation, so our formula becomes:
gh = 0.5v^2
Now, we just have to solve for v by plugging in the given values:
(9.8 m/s^2) * (2.41 m) = 0.5 * v^2
Tapping some numbers into the calculator, we get:
23.938 m^2/s^2 = 0.5 * v^2
Now, let's take the square root of both sides:
√(23.938 m^2/s^2) = v
Drumroll please...
v = 4.89 m/s
So, the swimmer's speed at the bottom of the slide will be approximately 4.89 meters per second. Hold onto your swimsuits, it's gonna be a splashy ride!
To find the swimmer's speed at the bottom of the slide, we can use the principle of conservation of energy.
According to this principle, the initial potential energy (PE) of the swimmer is converted into kinetic energy (KE) at the bottom of the slide.
The potential energy can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the swimmer
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = vertical height of the slide (2.41 m)
Assuming the swimmer's mass is 1 kg, we can calculate the potential energy:
PE = 1 kg * 9.8 m/s^2 * 2.41 m
PE = 23.758 Joules
Since there is no friction, the swimmer's potential energy is fully converted into kinetic energy at the bottom of the slide.
The kinetic energy can be calculated using the formula:
KE = 0.5 * m * v^2
Where:
v = velocity (speed) at the bottom of the slide
Setting the potential energy equal to the kinetic energy:
PE = KE
23.758 J = 0.5 * 1 kg * v^2
Simplifying the equation:
23.758 J = 0.5 kg * v^2
47.516 J/kg = v^2
Taking the square root of both sides:
v ≈ √(47.516 J/kg)
v ≈ 6.901 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 6.901 m/s.
To find the swimmer's speed at the bottom of the slide, we can use the principle of conservation of energy.
First, let's calculate the potential energy at the top of the slide. We can use the formula:
Potential Energy = mass * gravity * height
Since the swimmer starts at rest, her initial kinetic energy is zero. Therefore, all the initial potential energy will be converted into kinetic energy at the bottom of the slide. Hence, we can equate the potential energy at the top of the slide to the kinetic energy at the bottom.
Potential Energy at the top = Kinetic Energy at the bottom
So,
mass * gravity * height = 1/2 * mass * velocity^2
We can cancel out the mass from both sides of the equation.
gravity * height = 1/2 * velocity^2
Now, let's rearrange the equation to solve for velocity.
velocity^2 = 2 * gravity * height
velocity = sqrt(2 * gravity * height)
Given that the height of the slide is 2.41 m and the acceleration due to gravity is approximately 9.8 m/s^2, we can substitute these values into the equation to calculate the swimmer's speed at the bottom of the slide.
velocity = sqrt(2 * 9.8 * 2.41)
velocity = sqrt(47.924)
velocity ≈ 6.92 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 6.92 m/s.