At an amusement park, a swimmer uses a water slide to enter the main pool. The swimmer slides without friction, and descends through a vertical height of 2.89 m. Find the swimmer's speed at the bottom of the slide if she starts with an initial speed of 0.870 m/s.
Intial PE+ Initial KE= final KE
To find the swimmer's speed at the bottom of the slide, we can make use of the principle of conservation of energy. The total mechanical energy of the swimmer (potential energy + kinetic energy) remains constant throughout the slide.
The initial mechanical energy (E1) of the swimmer at the top of the slide is given by the sum of potential energy and kinetic energy:
E1 = mgh + 0.5mv^2
where:
m = mass of the swimmer
g = acceleration due to gravity (9.8 m/s^2)
h = vertical height of the slide
v = initial speed of the swimmer
Since the swimmer starts with an initial speed of 0.870 m/s, the initial mechanical energy can be written as:
E1 = mgh + 0.5m(0.870)^2
At the bottom of the slide, the potential energy is zero since the swimmer is at ground level. Therefore, the final mechanical energy (E2) is given by the kinetic energy:
E2 = 0.5mv^2_f
where:
v_f = final speed of the swimmer at the bottom of the slide
According to the principle of conservation of energy, E1 = E2:
mgh + 0.5m(0.870)^2 = 0.5mv^2_f
We can simplify the equation by canceling out the mass (m) on both sides:
gh + 0.5(0.870)^2 = 0.5v^2_f
Now we can solve for v_f, the final speed of the swimmer at the bottom of the slide:
0.5(0.870)^2 = 0.5v^2_f - gh
Multiply both sides by 2 to isolate v^2_f:
(0.870)^2 = v^2_f - 2gh
Now plug in the known values and solve for v_f:
(0.870)^2 = v^2_f - 2(9.8)(2.89)
Simplifying:
(0.760) = v^2_f - 56.744
Rearranging the equation:
v^2_f = 56.744 + 0.760
v^2_f = 57.504
Finally, take the square root of both sides to find v_f:
v_f = √57.504
v_f ≈ 7.58 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 7.58 m/s.