A child riding a bicycle has a total mass of 40.0 kg. The child approaches the top of a frictionless hill that is 10.0 m high and 100.0 m long at 5.0 m/s. What is the child’s velocity at the bottom of the hill? b) (Disregard air resistance. g = 9.8 m/s2.)
According to the work energy theorem,
Initial (KE+PE) =Final (KE+PE)
0.5x40x5^2 + 40x9.8x10 = KE final + 0
500 + 3920 = 1/2 mv2^2
v2^2 = 2 x 4420 / 40
v2 = 14.87 m/s
Frictionless hill is considered.
1/2 mv^2 = mgh
To find the child's velocity at the bottom of the hill, we can use the principle of conservation of energy. According to this principle, the total mechanical energy of a system remains constant when only conservative forces are acting on it.
At the top of the hill, the child has potential energy due to its height above the ground. At the bottom of the hill, all of the potential energy is converted to kinetic energy (energy of motion).
1. Calculate the potential energy at the top of the hill:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
PE = 40.0 kg * 9.8 m/s² * 10.0 m
2. Calculate the kinetic energy at the bottom of the hill:
Since no energy is lost due to friction or air resistance, the potential energy at the top of the hill is converted to kinetic energy at the bottom.
Kinetic energy (KE) = Potential energy at the top of the hill
KE = 40.0 kg * 9.8 m/s² * 10.0 m
3. Use the formula for kinetic energy:
KE = 0.5 * mass (m) * velocity squared (v²)
40.0 kg * 9.8 m/s² * 10.0 m = 0.5 * 40.0 kg * v²
4. Calculate the velocity at the bottom of the hill:
Rearrange the equation to solve for velocity (v):
v² = (40.0 kg * 9.8 m/s² * 10.0 m) / (0.5 * 40.0 kg)
v² = 9.8 m/s² * 10.0 m / 0.5
v² = 196 m²/s²
v = √(196 m²/s²)
v ≈ 14 m/s
Therefore, the child's velocity at the bottom of the hill is approximately 14 m/s.