the height above the ground of a child on a swing varies from 0.5m at the lowest point to 2.0m at the highest point. what is the maximum speed of the child?
5.4m/s
To find the maximum speed of the child on a swing, we can use the principle of conservation of mechanical energy. The total mechanical energy of the child on the swing remains constant throughout the motion.
The total mechanical energy (E) is the sum of the potential energy (PE) and the kinetic energy (KE). At the highest point, all the energy is in the form of potential energy. At the lowest point, all the energy is in the form of kinetic energy.
Given:
Height at the lowest point, h_lowest = 0.5m
Height at the highest point, h_highest = 2.0m
We can calculate the maximum speed using the equation:
E_lowest = E_highest
PE_lowest + KE_lowest = PE_highest + KE_highest
At the lowest point:
PE_lowest = m * g * h_lowest (where m is the mass and g is the acceleration due to gravity)
KE_lowest = 0 (as the velocity is zero at the lowest point)
At the highest point:
PE_highest = m * g * h_highest
KE_highest = 0.5 * m * v^2 (where v is the maximum speed)
Setting up the equation:
m * g * h_lowest = m * g * h_highest + 0.5 * m * v^2
Simplifying the equation:
g * h_lowest = g * h_highest + 0.5 * v^2
Solving for v (maximum speed):
0.5 * v^2 = g * (h_lowest - h_highest)
v^2 = 2 * g * (h_lowest - h_highest)
v = √(2 * g * (h_lowest - h_highest))
Substituting the given values:
v = √(2 * 9.8 * (0.5 - 2.0))
Calculating the maximum speed:
v = √(2 * 9.8 * (-1.5))
v = √(-29.4)
v ≈ 5.42 m/s
Therefore, the maximum speed of the child on the swing is approximately 5.42 m/s.
To find the maximum speed of the child on the swing, we can use the principle of conservation of energy.
At the highest point, all of the potential energy is converted into kinetic energy. At the lowest point, all of the kinetic energy is converted into potential energy. Since the swing is assumed to be frictionless, we can ignore any energy losses.
The potential energy at the highest point is given by the formula:
P.E.(highest) = m * g * h
where m is the mass of the child, g is the acceleration due to gravity (9.8 m/s²), and h is the height above the ground at the highest point (2.0 m).
The kinetic energy at the lowest point is given by the formula:
K.E.(lowest) = 0.5 * m * v²
where v is the velocity at the lowest point.
Since the potential energy at the highest point is equal to the kinetic energy at the lowest point, we can set the two equations equal to each other:
m * g * h = 0.5 * m * v²
Simplifying the equation, we can cancel out the mass 'm':
g * h = 0.5 * v²
Now we can solve for the velocity 'v':
v² = 2 * g * h
v = √(2 * g * h)
Substituting the values, we get:
v = √(2 * 9.8 * 2.0)
v = √(39.2)
v ≈ 6.26 m/s
Therefore, the maximum speed of the child on the swing is approximately 6.26 m/s.