Final answer:
To find the height the toboggan will reach, we can use the conservation of mechanical energy. By converting the initial kinetic energy of the toboggan into gravitational potential energy, we can solve for the height using the equation for potential energy. Plugging in the given values, we find that the toboggan will reach a height of 3.61 meters above the base before stopping.
Step-by-step explanation:
To solve this problem, we can use the conservation of mechanical energy. At the base of the hill, the toboggan has kinetic energy given by KE₁ = 1/2 mv₁², where m is the mass of the toboggan and v₁ is its initial speed. As the toboggan goes up the hill, its gravitational potential energy increases and its kinetic energy decreases. At the top of the hill, when the toboggan comes to a stop, all of its initial kinetic energy has been converted into gravitational potential energy. Using the equation for gravitational potential energy, PE = mgh, where g is the acceleration due to gravity and h is the height above the base, we can solve for h.
First, we need to find the component of the toboggan's initial velocity that is directed up the hill. Since the hill makes an angle of 25.0 degrees with the horizontal, the component of the initial velocity along the hill is given by v₁sin(25.0°). We can then use this value to find the height above the base that the toboggan will reach.
h = (v₁sin(25.0°))² / (2g)
Plugging in the given values of v₁ = 12.5 m/s and the acceleration due to gravity g = 9.8 m/s², we can calculate the height h.
h = (12.5sin(25.0°))² / (2 * 9.8) = 3.61 m
Therefore, the toboggan will go vertically 3.61 meters above the base before stopping.