A girl swings on a playground swing in such a way that at her highest point she is 4.1 m from the ground, while at her lowest point she is 0.8 m from the ground.What is her maximum speed? The acceleration due to gravity is 9.8 m/s2 .Answer in units of m/s.
At the highest point of her swing, her kinetic energy is converted into potential energy, and at the lowest point, her potential energy is converted into kinetic energy. We can use the conservation of mechanical energy to find her maximum speed at the lowest point.
At the highest point:
Potential energy (PE_h) = m * g * h_h
At the lowest point:
Kinetic energy (KE_l) = 0.5 * m * v^2
Potential energy (PE_l) = m * g * h_l
Conservation of mechanical energy:
PE_h = KE_l + PE_l
Since we only need to find her maximum speed, we don't need to worry about the mass (m) as it will cancel out.
Substituting the values:
(9.8 m/s²)(4.1 m) = 0.5(v^2) + (9.8 m/s²)(0.8 m)
Solving for v:
(9.8)(4.1-0.8) = 0.5(v^2)
9.8 * 3.3 = 0.5(v^2)
v^2 = (9.8 * 3.3) / 0.5
v^2 = 64.34
v = √64.34
v ≈ 8.02 m/s
The girl's maximum speed is approximately 8.02 m/s.
To find the maximum speed of the girl on the swing, we need to use the conservation of energy principle.
At the highest point, all of the girl's energy is potential energy, while at the lowest point, all of her energy is kinetic energy.
The potential energy (PE) can be calculated using the formula: PE = m * g * h, where m is the girl's mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
At the highest point, the potential energy is given by: PE_highest = m * g * h_highest = m * 9.8 * 4.1
At the lowest point, all the potential energy has converted to kinetic energy, which is given by: KE_lowest = (1/2) * m * v^2, where v is the velocity.
Equating the potential and kinetic energy, we get: PE_highest = KE_lowest
m * 9.8 * 4.1 = (1/2) * m * v^2
Canceling out the mass on both sides, we get:
9.8 * 4.1 = (1/2) * v^2
Simplifying the equation, we have:
v^2 = (9.8 * 4.1) / 0.5
v^2 = 40.18
Taking the square root of both sides to solve for v, we get:
v = √40.18
v ≈ 6.34 m/s
Therefore, the maximum speed of the girl on the swing is approximately 6.34 m/s.
To find the girl's maximum speed on the swing, we can use the principle of conservation of mechanical energy. At her highest point, all her initial gravitational potential energy is converted into kinetic energy. Similarly, at her lowest point, all her kinetic energy is converted back into gravitational potential energy.
The formula for gravitational potential energy is given by:
Potential Energy = m * g * h
where m is the girl's mass, g is the acceleration due to gravity, and h is the height.
Since the girl's mass is not given, we can ignore it in this calculation because it cancels out when finding the maximum speed.
At her highest point:
Potential Energy = Kinetic Energy
m * g * h = 1/2 * m * v^2
where v is the maximum speed we're trying to find.
We can rearrange the equation to solve for v:
v^2 = 2 * g * h
v = √(2 * g * h)
Now we can substitute the given values:
g = 9.8 m/s^2
h = 4.1 m
v = √(2 * 9.8 * 4.1)
v = √(80.36)
v ≈ 8.97 m/s
Therefore, the girl's maximum speed on the swing is approximately 8.97 m/s.