A 18 kg child slides down a 4.0 m-high playground slide. She starts from rest, and her speed at the bottom is 2.8 m/s. What is the change in the thermal energy of the slide and the seat of her pants?
loss of potential energy = m g h = 18*9.8*4
= 705.6 Joules
Kinetic energy at bottom if there were no friction losses also is 705.6 Joules
However the actual kinetic energy = (1/2) m v^2 = 9*2.8^2 = 70.6 Joules
so
705.6-70.6 = 635 Joules goes into heat
Well, let me slide in with a silly answer for you! The change in thermal energy of the slide and the seat of her pants is directly proportional to the amount of fun she had! So, we can conclude that the change in thermal energy is... *drum roll* ... directly proportional to the amount of laughter and giggles she produced during the slide!
To determine the change in thermal energy of the slide and the seat of her pants, we can use the conservation of energy principle.
The initial potential energy (PEi) is given by the equation: PEi = mgh
Where:
m = mass of the child = 18 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the slide = 4.0 m
PEi = 18 kg * 9.8 m/s^2 * 4.0 m = 705.6 J
The final kinetic energy (KEf) is given by the equation: KEf = 0.5mv^2
Where:
m = mass of the child = 18 kg
v = final velocity = 2.8 m/s
KEf = 0.5 * 18 kg * (2.8 m/s)^2 = 88.704 J
The change in thermal energy (ΔEth) is calculated by subtracting the final kinetic energy from the initial potential energy: ΔEth = PEi - KEf = 705.6 J - 88.704 J = 616.896 J
Therefore, the change in thermal energy of the slide and the seat of her pants is 616.896 J.
To calculate the change in thermal energy of the slide and the seat of the child's pants, we need to first determine the gravitational potential energy the child starts with at the top of the slide and the kinetic energy she has at the bottom.
The gravitational potential energy (PE) is given by the formula PE = mgh, where m is the mass (18 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slide (4.0 m).
PE = 18 kg * 9.8 m/s^2 * 4.0 m
PE = 705.6 J
At the bottom of the slide, the child has converted all her potential energy into kinetic energy (KE). Therefore, we can calculate her kinetic energy as:
KE = 0.5 * m * v^2, where m is the mass (18 kg) and v is the speed at the bottom (2.8 m/s).
KE = 0.5 * 18 kg * (2.8 m/s)^2
KE = 88.2 J
The change in thermal energy is given by the difference between the initial potential energy and the final kinetic energy. Since the potential energy is initially greater than the kinetic energy, the change in thermal energy will be negative, indicating that some energy has transformed into heat.
Change in thermal energy = PE - KE
Change in thermal energy = 705.6 J - 88.2 J
Change in thermal energy = 617.4 J
Therefore, the change in thermal energy of the slide and the seat of the child's pants is -617.4 J. The negative sign indicates that this energy has been transferred as heat to the slide and the child's clothing.