A student, starting from rest, slides down a water slide. On the way down, a kinetic frictional force (a nonconservative force) acts on her. The student has a mass of 82 kg, and the height of the water slide is 11.3 m. If the kinetic frictional force does -6.2 × 103 J of work, how fast is the student going at the bottom of the slide? Use g = 9.81 m/s2
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To find the speed of the student at the bottom of the slide, we can use the principle of conservation of mechanical energy.
The conservation of mechanical energy states that the total mechanical energy of a system remains constant if no external forces (except non-conservative forces like friction) act on it. Therefore, the initial potential energy of the student at the top of the slide will be equal to the final kinetic energy of the student at the bottom.
The potential energy (PE) of an object at a certain height h is given by the formula:
PE = mgh
Where:
m = mass of the object
g = acceleration due to gravity (9.81 m/s^2)
h = height
So, the potential energy of the student at the top of the slide is:
PE = (82 kg) × (9.81 m/s^2) × (11.3 m)
Now, since there is a kinetic frictional force acting on the student, which does negative work, the total mechanical energy of the system will decrease. The work done by the kinetic frictional force is equal to the change in mechanical energy:
Work (W) = Change in mechanical energy
In this case, the work done by the kinetic frictional force is given as -6.2 × 10^3 J. So:
W = -6.2 × 10^3 J
Now, at the bottom of the slide, when the student reaches the ground, all the potential energy is converted into kinetic energy. The kinetic energy (KE) of an object is given by the formula:
KE = (1/2) mv^2
Where:
m = mass of the object
v = velocity or speed
Since we're trying to find the speed of the student, we'll set the initial and final potential energy equal to the final kinetic energy:
PE - W = KE
(mgh) - W = (1/2) mv^2
(82 kg) × (9.81 m/s^2) × (11.3 m) - (-6.2 × 10^3 J) = (1/2) (82 kg) × v^2
Solving for v, we can rearrange the equation:
v^2 = [(82 kg) × (9.81 m/s^2) × (11.3 m) - (-6.2 × 10^3 J)] / (1/2) (82 kg)
Now, we can simply calculate v by taking the square root of both sides of the equation:
v = √[[ (82 kg) × (9.81 m/s^2) × (11.3 m) - (-6.2 × 10^3 J)] / (1/2) (82 kg)]
Evaluating this expression will give us the speed at the bottom of the slide.