Final answer:
a. The sled reaches a vertical height of 5.10 m above its starting point. b. The sled's speed when it slides back down to its starting point is 10 m/s.
Step-by-step explanation:
a. To find the vertical height the sled reaches above its starting point, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the sled is equal to the sum of its potential energy at the highest point and its final kinetic energy. The initial kinetic energy is given by:
KE = 1/2 * m * v^2
where m is the mass of the sled and v is its initial speed.
The potential energy at the highest point is given by:
PE = m * g * h
where g is the acceleration due to gravity and h is the vertical height above the starting point.
Since the sled comes to a stop at the highest point, its final kinetic energy is zero. Therefore, we can set up the equation:
1/2 * m * v^2 = m * g * h
Simplifying and solving for h, we get:
h = v^2 / (2 * g)
Plugging in the given values, we have:
h = (10 m/s)^2 / (2 * 9.8 m/s^2) = 5.10 m
So, the sled reaches a vertical height of 5.10 m above its starting point.
b. To find the sled's speed when it slides back down to its starting point, we can use the principle of conservation of mechanical energy again. The potential energy at the highest point is converted back into kinetic energy as the sled slides back down. Therefore, we can set up the equation:
1/2 * m * v^2 = m * g * h
where m is the mass of the sled, v is its final velocity, g is the acceleration due to gravity, and h is the vertical height above the starting point.
Since the height is the same as before (5.10 m) and the mass of the sled is still 5.0 kg, we can rearrange the equation to solve for v:
v = sqrt(2 * g * h)
Plugging in the given values, we have:
v = sqrt(2 * 9.8 m/s^2 * 5.10 m) = 10 m/s
So, the sled's speed when it slides back down to its starting point is 10 m/s.