At a playground, a 19.8 kg child plays on a slide that drops through a height of 2.13 m. The child starts at rest at the top of the slide. On the way down, the slide does a nonconservative work of -319 J on the child. What is the child's speed at the bottom of the slide?
To find the child's speed at the bottom of the slide, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the system (child) is the sum of its potential energy (PE) and kinetic energy (KE). According to the conservation of mechanical energy, the total mechanical energy of the system is conserved, meaning it remains constant throughout the motion.
Mathematically, we can express this principle as:
PE_initial + KE_initial + Work_nonconservative = PE_final + KE_final
Since the child starts at rest at the top of the slide, the initial kinetic energy (KE_initial) is zero. Also, at the bottom of the slide, the child will have zero potential energy (PE_final) but nonzero kinetic energy (KE_final).
The work done by the nonconservative force on the child is given as -319 J, indicating work is done against the motion of the child.
Applying the conservation of mechanical energy equation, we have:
0 + 0 + (-319 J) = 0 + (1/2)mv^2
Where:
m = mass of the child = 19.8 kg
v = speed of the child at the bottom of the slide (what we need to find)
Simplifying the equation:
-319 J = (1/2)(19.8 kg)v^2
To isolate v^2, divide both sides by (1/2)(19.8 kg):
(-319 J) / (1/2)(19.8 kg) = v^2
Now, calculate the left side:
(-319 J) / (1/2)(19.8 kg) ≈ -32.1162 m^2/s^2
Taking the square root of both sides, we get:
v ≈ √(-32.1162 m^2/s^2)
Since the square root of a negative number is imaginary, it means that the child cannot reach that speed. In this case, due to the negative work done by the nonconservative force, the child will slow down and come to a stop before reaching the bottom of the slide.