how much energy was converted to Thermal Energy

Question

The "Niagara Falls" water slide to waterpark and has a height of 76.0 feet (23.2 meters). Because the slide is not completely vertical the total length of the slide is 90.0 feet (27.4 meters). A 70.0 kg person slides down this slide starting from rest.

a) If the slide is frictionless, what should the person's velocity be at the bottom of the slide in meters/second?

b) What is the velocity in miles per hour?

c)What if there was friction between the slide and the person. If the velocity of the person at the bottom of the slide was really 38.2 mi/hour how much energy was converted to Thermal Energy in this problem?

d) From what you founded above, What must the magnitude of the friction force have been? (friction force remained the same and constant)

Please solve it please!

Answer 1

Given:

height (h) = 23.2 m

mass (m) = 70 kg

frictionless

A) To solve for (a) I used the principal of the Conservation of Mechanical energy:

mgh = (1/2)mv²

(70kg)(9.81 m/s²)(23.3m) = (1/2)(70kg)(v²)

(70kg)(9.81 m/s²)(23.3m) = (1/2)(70kg)(v²)

Solve for v: v² = 455.184 m²/s²

v = 21.34 m/s

B) To convert to miles an hour is easy so long as you keep track of the units!

1 mile = 1.609km = 1609.34m

1 hour = 3600 seconds

21.34 m/s (1 mile/1609.34 m)*(3600 s/1 hr)

21.34 m/s (1 mile/1609.34 m)*(3600 s/1 hr) *Check that the units match mi/hr

v = 47.73 mi/hr

C) Since there is friction we can automatically assume that there is a loss in both exit velocity and energy in the form of heat (thermal loss). I use the equation that relates thermal energy to the difference in potential and kinetic energy. For question (a) TE was assumed to be 0 with a frictionless slide so PE = KE. This time is different. Also we need to convert 38.2 mi/hr into m/s.

38.2 mi/hr (1609.34m / 1mi)*(1hr/ 3600s)

38.2 mi/hr (1609.34m / 1mi)*(1hr/ 3600s) *Check that answer is in m/s

v = 17.0769 m/s

TE = PE - KE

TE = mgh - (1/2)mv²

TE = (70kg)(9.81m/s²)(23.2m) - (1/2)(70kg)(17.0769 m/s)²

TE = 5724.72 kg m²/s² * 1kg m²/s² = 1 Joule

TE = 5724.72 Joules (J)

D) Magnitude infers that there is a force AND a direction. Since friction is opposite that of movement you just have to draw out the triangle and solve for that angle!

*x is the angle sin(x) = o/h

sin(x) = 23.2m / 27.4m

x = sin^-1(23.2 / 27.4)

x = 57.856*

Now we need to assume that the acceleration is constant since the friction would really change along the length of the slide. We assume that the acceleration is constant.

***If this assumption is wrong then so is the answer!***

f_k = u_K (mg*cos(x))

where u_K = tan(x) which is the kinetic friction coefficient

f_D = 581.438 N

Hope this helps!