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 decends through a vertical height of 2.40 m, what is her speed at the bottom of the slide?
The use of conservation of energy approach will tell you right away that
V = sqrt(2 g H)
loss of potential energy = M g H = gain in kinetic energy = (1/2) M V^2
Therefore 2gH = V^2
V = sqrt(2 g H)
Well, I must say, this swimmer is really going with the flow! Let's calculate her speed at the bottom of the slide.
To find the speed, we can use the principle of conservation of energy. At the top of the slide, the swimmer has only potential energy, which is converted into kinetic energy at the bottom of the slide.
The potential energy at the top is given by mgh, where m is the mass (which isn't given), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the slide (2.40 m).
Since the swimmer starts at rest, her initial kinetic energy is zero. The total mechanical energy (potential + kinetic) is conserved, so we have:
mgh = 1/2 mv²
This equation shows that mass cancels out. So, the swimmer's speed at the bottom of the slide doesn't depend on her mass.
Therefore, her speed at the bottom of the slide is the square root of (2gh).
Now, let's crunch the numbers. Inserting the values, we get:
v = √(2 * 9.8 m/s² * 2.40 m)
After simplifying, we find:
v ≈ 6.94 m/s
So, the swimmer will be making quite the splash at the bottom of the slide with a speed of approximately 6.94 m/s. Just be warned: high-speed belly flops are not recommended!
To find the swimmer's speed at the bottom of the slide, we can use the principle of conservation of energy.
According to the law of conservation of energy, the total mechanical energy of a system remains constant as long as no external forces, such as friction or air resistance, are acting on it. In this case, since there is no mention of any external forces, we can assume that mechanical energy is conserved throughout the slide.
The total mechanical energy of the system consists of the potential energy (PE) and the kinetic energy (KE) of the swimmer. At the top of the slide, when the swimmer is at rest, all her energy is in the form of potential energy. As she slides down, her potential energy is converted into kinetic energy.
The potential energy (PE) of an object is given by the formula:
PE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
At the top of the slide, the potential energy is maximum, and at the bottom of the slide, it is minimum (zero). Therefore, we can equate the initial potential energy (PEi) to the final kinetic energy (KEf).
PEi = KEf
m * g * h = 1/2 * m * v^2
where v is the velocity (speed) of the swimmer at the bottom of the slide.
We can cancel out the mass (m) on both sides of the equation, leaving us with:
g * h = 1/2 * v^2
Now, we can rearrange the equation to solve for v:
v^2 = 2 * g * h
Taking the square root of both sides, we get:
v = √(2 * g * h)
Plugging in the values given in the problem:
v = √(2 * 9.8 m/s^2 * 2.40 m)
Simplifying:
v = √(19.6 m^2/s^2 * 2.40 m)
v = √(47.04 m^3/s^2)
Taking the square root:
v ≈ √47.04 m/s
v ≈ 6.86 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 6.86 m/s.