A child riding a bicycle has a total mass of 40kg. The child approaches the top of a hill that is 10m high and 100m long at 5m/s. If the force of friction between the bicycle and the hill is 20N, what is the child's velocity at the bottom of the hill? (Assume no air resistance and that g=9.81m/s)
Well, this is quite a hill-arious question! Let's break it down step by step.
First, we need to find the potential energy of the child at the top of the hill. The formula for potential energy is PE = mgh, where m is the mass, g is the gravitational acceleration (9.81 m/s^2), and h is the height. So, PE = 40 kg * 9.81 m/s^2 * 10 m = 3924 J.
Now, as the child goes down the hill, the potential energy gets converted into kinetic energy. The formula for kinetic energy is KE = 1/2 mv^2, where m is the mass and v is the velocity. Since there is no air resistance, all of the potential energy will be converted into kinetic energy, neglecting any loss due to friction.
So, KE = 3924 J = 1/2 * 40 kg * v^2. Solving for v, we get v^2 = (2 * 3924 J) / 40 kg = 196.2. Taking the square root of both sides, we find v ≈ 14 m/s.
But wait! We also have to consider the force of friction. The work done by friction will be equal to the force of friction times the distance traveled. Since work is equal to force times distance, we have W = Fd. Plugging in the given values, W = 20 N * 100 m = 2000 J.
So, the kinetic energy at the bottom of the hill will be slightly less than 3924 J due to the work done by friction. Let's call it KE_f to differentiate it from the initial kinetic energy. KE_f = KE - W = 3924 J - 2000 J = 1924 J.
Now we can solve for the final velocity. KE_f = 1/2 mv_f^2, so 1924 J = 1/2 * 40 kg * v_f^2. Solving for v_f, we find v_f^2 = (2 * 1924 J) / 40 kg = 96.2. Taking the square root of both sides, we get v_f ≈ 9.8 m/s.
So, the child's velocity at the bottom of the hill, taking friction into account, is approximately 9.8 m/s. Keep on rolling and have fun calculating!
To determine the child's velocity at the bottom of the hill, we can use the principle of conservation of energy.
1. First, find the potential energy at the top of the hill:
Potential energy = mass * gravitational acceleration * height
Potential energy = 40 kg * 9.81 m/s^2 * 10 m
Potential energy = 3924 J
2. Next, find the kinetic energy at the top of the hill:
Kinetic energy = 0.5 * mass * velocity^2
Kinetic energy = 0.5 * 40 kg * (5 m/s)^2
Kinetic energy = 500 J
3. Since there is no air resistance, the total mechanical energy (sum of potential and kinetic energy) is conserved.
Total mechanical energy at the top = Total mechanical energy at the bottom
Potential energy at the top + Kinetic energy at the top = Potential energy at the bottom + Kinetic energy at the bottom
(Potential energy + Kinetic energy) at the top = (Potential energy + Kinetic energy) at the bottom
3924 J + 500 J = 40 kg * 9.81 m/s^2 * 0 m + Kinetic energy at the bottom
4424 J = Kinetic energy at the bottom
4. Now we can find the velocity at the bottom of the hill using the kinetic energy formula:
Kinetic energy = 0.5 * mass * velocity^2
4424 J = 0.5 * 40 kg * velocity^2
Simplifying the equation:
4424 J = 20 kg * velocity^2
velocity^2 = 4424 J / 20 kg
velocity^2 = 221.2 m^2/s^2
Taking the square root of both sides to find the velocity:
velocity = √(221.2 m^2/s^2)
velocity ≈ 14.88 m/s
Therefore, the child's velocity at the bottom of the hill is approximately 14.88 m/s.
To find the child's velocity at the bottom of the hill, we need to consider the conservation of energy.
The initial energy of the child is given by the sum of their kinetic energy (KE) and potential energy (PE) at the top of the hill:
Initial Energy = KE + PE
The kinetic energy is given by the equation:
KE = (1/2) * mass * velocity^2
The potential energy is given by the equation:
PE = mass * gravity * height
Given:
- Mass of the child (m) = 40 kg
- Height of the hill (h) = 10 m
- Velocity at the top of the hill (v) = 5 m/s
- Force of friction (F) = 20 N
- Acceleration due to gravity (g) = 9.81 m/s^2
First, we need to calculate the initial energy:
Initial Energy = (1/2) * mass * velocity^2 + mass * gravity * height
Substituting the given values:
Initial Energy = (1/2) * 40 kg * (5 m/s)^2 + 40 kg * 9.81 m/s^2 * 10 m
Now, we need to consider the work done against friction. The work done is equal to the force of friction multiplied by the distance traveled:
Work = F * d
Since the force of friction is given as 20 N and the distance traveled is 100 m, we have:
Work = 20 N * 100 m
The work done against friction is negative because it acts in the opposite direction to the motion.
Next, we need to consider the final energy at the bottom of the hill. At the bottom, all the initial potential energy is converted to kinetic energy:
Final Energy = (1/2) * mass * velocity^2
We can now equate the initial energy to the final energy and solve for the velocity at the bottom of the hill:
Initial Energy - Work = Final Energy
((1/2) * 40 kg * (5 m/s)^2 + 40 kg * 9.81 m/s^2 * 10 m) - (20 N * 100 m) = (1/2) * 40 kg * velocity^2
Simplifying the equation:
(10,000 J + 3,924 J) - 2,000 J = 20 kg * velocity^2
13,924 J - 2,000 J = 20 kg * velocity^2
11,924 J = 20 kg * velocity^2
Divide both sides by 20 kg:
velocity^2 = 11,924 J / 20 kg
velocity^2 = 596.2 m^2/s^2
Take the square root of both sides to find velocity:
velocity = √(596.2 m^2/s^2)
velocity ≈ 24.4 m/s
Therefore, the child's velocity at the bottom of the hill is approximately 24.4 m/s.
Energy used up by friction:20N(100m)
energy gained by bike: 40*9.8*10
Ke at bottom=energy gained minus friction
veloctiy at bottom=sqrt(2KE/mass)