You're standing at the top of a snow-covered hill, sloped at 11.0°. The coefficient of kinetic friction between your sled and the hill is 0.304. Unfortunately, there's a running stream just at the bottom of the hill, and you don't want to get wet. If you and your sled together have a mass of 83.0 kg, with what speed should you push off with to stop just before the stream, 97.0 m down the slope?
the work done by friction must equal the
... initial kinetic energy plus the gravitational energy from the height change
f * d = (1/2 m v^2) + (m g h)
To find the speed at which you should push off in order to stop just before the stream, we can use the concept of work and energy. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
First, let's calculate the gravitational potential energy (PE) at the top of the slope using the mass (m) and the height (h):
PE = m * g * h
where:
m = mass of the object (83.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the slope (sin(11.0°) * 97.0 m)
Next, let's calculate the kinetic energy (KE) at the bottom of the slope:
KE = 0.5 * m * v^2
where:
v = velocity of the sled at the bottom of the slope
Since there is no change in the sled's mechanical energy (work done against friction), the initial gravitational potential energy should be equal to the final kinetic energy.
PE = KE
m * g * h = 0.5 * m * v^2
Now, we need to isolate the velocity (v) in the equation:
v^2 = (2 * g * h)
v = √(2 * g * h)
Substituting the given values:
v = √(2 * 9.8 m/s^2 * sin(11.0°) * 97.0 m)
Simplifying this equation will give you the speed at which you should push off to stop just before the stream.