Barney and his friends are on a toboggan (with a combined mass of 95 kg) at the top of a 25 m hill. They start down the hill with an initial velocity of 3 m/s What is their starting KE? What is their starting PE? What is their total Mechanical Energy?How fast are they moving at the bottom of the hill?

Here's how to solve this problem:<br /><br />**1. Calculate Starting Kinetic Energy (KE):**<br /><br />* Formula: KE = (1/2) * m * v²<br />* m (mass) = 95 kg<br />* v (velocity) = 3 m/s<br />* KE = (1/2) * 95 kg * (3 m/s)²<br />* KE = 427.5 Joules<br /><br />**2. Calculate Starting Potential Energy (PE):**<br /><br />* Formula: PE = m * g * h<br />* m (mass) = 95 kg<br />* g (acceleration due to gravity) = 9.8 m/s²<br />* h (height) = 25 m<br />* PE = 95 kg * 9.8 m/s² * 25 m<br />* PE = 23275 Joules<br /><br />**3. Calculate Total Mechanical Energy:**<br /><br />* Total Mechanical Energy = KE + PE<br />* Total Mechanical Energy = 427.5 Joules + 23275 Joules<br />* Total Mechanical Energy = 23702.5 Joules<br /><br />**4. Calculate Velocity at the Bottom of the Hill:**<br /><br />* At the bottom of the hill, all the potential energy is converted into kinetic energy. We can ignore friction in this problem. Therefore, the total mechanical energy at the bottom of the hill is equal to the kinetic energy at the bottom of the hill.<br />* Total Mechanical Energy at the bottom = KE at the bottom<br />* 23702.5 Joules = (1/2) * m * v²<br />* 23702.5 Joules = (1/2) * 95 kg * v²<br />* Solving for v:<br /> * v² = (2 * 23702.5 Joules) / 95 kg<br /> * v² = 498.99 m²/s²<br /> * v = √498.99 m²/s²<br /> * v ≈ 22.34 m/s<br /><br />**Answer:**<br /><br />* Starting KE: 427.5 Joules<br />* Starting PE: 23275 Joules<br />* Total Mechanical Energy: 23702.5 Joules<br />* Velocity at the bottom of the hill: Approximately 22.34 m/s<br />