Starting at rest, Tina slides down a frictionless waterslide with a horizontal section at the bottom that is 6.10 ft above the surface of the swimming pool and strikes the water a distance of 13.2 ft away from the end of the slide. Using conservation of energy, what is Tina's initial height on the waterslide?
[ ] ft above the bottom of the slide
How long does it take to fall 6.10 ft?
16t^2 = 16.10
Now, knowing t, v at the bottom of the slide is 13.2/t ft/s
the PE lost on the slide is all KE at the bottom. So,
g(h-6.10) = 1/2 v^2
To determine Tina's initial height on the waterslide, we can use the principle of conservation of energy.
The initial potential energy (PEi) at the top of the waterslide is equal to the sum of the final potential energy (PEf) at the bottom of the slide and the final kinetic energy (KEf) when Tina reaches the water.
PEi = PEf + KEf
The potential energy (PE) is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Since the slide is frictionless, there is no loss of mechanical energy, and we can assume that the initial kinetic energy (KEi) is zero.
Therefore, we can rewrite the conservation of energy equation as:
PEi = PEf + KEf
mghi = mghf + KEf
Since the mass cancels out, we can simplify the equation to:
ghi = ghf + KEf
We know that the final kinetic energy is given by KEf = (1/2)mv^2, where m is the mass and v is the velocity.
Since Tina starts at rest, the final kinetic energy will be zero.
Therefore, the equation becomes:
ghi = ghf + 0
Simplifying further:
ghi = ghf
To find Tina's initial height on the waterslide, we need to determine the height at the bottom of the slide (hf), which is 6.10 ft above the surface of the swimming pool.
Therefore, Tina's initial height on the waterslide is:
ghi = hf + 6.10 ft above the bottom of the slide.
Answer: [ ] ft above the bottom of the slide, Tina's initial height on the waterslide is 6.10 ft above the bottom of the slide.
To find Tina's initial height on the waterslide, we can use the principle of conservation of energy. This principle states that the total mechanical energy of a system remains constant if only conservative forces (such as gravity) are acting on it.
Initially, Tina is at rest, so her initial mechanical energy consists only of gravitational potential energy. As she slides down the waterslide and reaches the horizontal section at the bottom, her potential energy is converted to kinetic energy.
The equation for gravitational potential energy is given by:
Potential Energy = mass * gravitational acceleration * height
Since the mass of Tina is not given in the problem, we can assume it cancels out during calculations.
At the top of the waterslide, Tina's mechanical energy is solely potential energy, given by:
Initial Energy = m * g * h
At the bottom of the slide, all of Tina's potential energy is converted to kinetic energy, given by:
Final Energy = 0.5 * m * v^2
where v is the velocity of Tina at the bottom of the slide.
Since energy is conserved, we can equate the initial energy to the final energy:
Initial Energy = Final Energy
m * g * h = 0.5 * m * v^2
We can cancel out the mass from both sides of the equation:
g * h = 0.5 * v^2
Now, we need to determine the velocity of Tina at the bottom of the slide. We can use kinematic equations to find this value.
Given that Tina slides a distance of 13.2 ft horizontally from the end of the slide to the point she strikes the water, and assuming no horizontal forces act on her, we can use the equation:
Distance = Velocity * Time
Since Tina starts from rest, her initial velocity is zero. The distance she slides horizontally is 13.2 ft, so we have:
13.2 ft = 0 * Time
This implies that the time it takes for Tina to slide horizontally is zero.
Now, let's consider Tina's vertical motion. We can use the equation of motion to determine her final vertical velocity. Since she starts from rest vertically, her initial vertical velocity is also zero. The equation for vertical displacement in terms of time and initial velocity is given by:
Vertical Displacement = Initial Vertical Velocity * Time + 0.5 * acceleration * Time^2
Since Tina strikes the water at a distance of 6.10 ft below the starting point (when measured vertically), we have:
-6.10 ft = 0 * Time + 0.5 * g * Time^2
Simplifying this equation:
6.10 ft = 0.5 * 32.2 ft/s^2 * Time^2
12.20 ft = 32.2 ft/s^2 * Time^2
Dividing both sides by 32.2 ft/s^2:
Time^2 = 12.20 ft / 32.2 ft/s^2
Time^2 ≈ 0.3795 s^2
Taking the square root of both sides:
Time ≈ √(0.3795 s^2)
Time ≈ 0.6166 s
Now, we can determine the final vertical velocity of Tina using the equation:
Final Vertical Velocity = Initial Vertical Velocity + acceleration * Time
Since her initial vertical velocity is zero, we have:
Final Vertical Velocity = g * Time
Final Vertical Velocity = 32.2 ft/s^2 * 0.6166 s
Final Vertical Velocity ≈ 19.83 ft/s
Finally, we can find Tina's initial height on the waterslide using the equation:
g * h = 0.5 * v^2
Solving for h:
h = (0.5 * v^2) / g
Substituting the values:
h = (0.5 * (19.83 ft/s)^2) / 32.2 ft/s^2
h ≈ 6.10 ft
Therefore, Tina's initial height on the waterslide is 6.10 ft above the bottom of the slide.