A child, starting from rest at the top of a
playground slide, reaches a speed of 7.0 meters
per second at the bottom of the slide. What is
the vertical height of the slide? [Neglect
friction.]
m g h = (1/2) m v^2
or
h = v^2/ (2g) = 49/(2*9.81) or about 2.5 meters
2.5 meters
To find the vertical height of the slide, we can use the conservation of energy. At the top of the slide, the total energy is in the form of gravitational potential energy, and at the bottom of the slide, it is in the form of kinetic energy.
The formula for gravitational potential energy is given by:
Potential Energy = mass * acceleration due to gravity * height
The formula for kinetic energy is given by:
Kinetic Energy = (1/2) * mass * velocity^2
Since the mass of the child is not specified and cancels out in the equation, we do not need it for our calculation.
Equating the potential energy at the top to the kinetic energy at the bottom, we have:
mass * acceleration due to gravity * height = (1/2) * mass * velocity^2
We can now cancel the mass on both sides of the equation:
acceleration due to gravity * height = (1/2) * velocity^2
Finally, we can solve for the height by rearranging the equation:
height = (1/2) * (velocity^2 / acceleration due to gravity)
Plugging in the values:
height = (1/2) * (7.0^2 / 9.8)
height = (1/2) * (49 / 9.8)
height = (1/2) * 5
height = 2.5 meters
Therefore, the vertical height of the slide is 2.5 meters.
To determine the vertical height of the slide, we can use the principles of conservation of energy. The total mechanical energy of the child at the top of the slide is equal to the total mechanical energy at the bottom of the slide.
At the top of the slide, the child has gravitational potential energy (PE) and no kinetic energy. As the child slides down, their potential energy is converted to kinetic energy. At the bottom of the slide, the child has no potential energy and only kinetic energy.
The equation for gravitational potential energy is: PE = m * g * h
Where:
- PE is the potential energy
- m is the mass of the child (which we can assume to be negligible)
- g is the acceleration due to gravity (approximated as 9.8 m/s²)
- h is the height of the slide
At the top of the slide, the child has zero kinetic energy. At the bottom of the slide, the child has kinetic energy, calculated by: KE = 0.5 * m * v²
Where:
- KE is the kinetic energy
- v is the velocity of the child at the bottom of the slide
Since the mechanical energy is conserved, we can set the potential energy at the top of the slide equal to the kinetic energy at the bottom of the slide:
PE = KE
Therefore,
m * g * h = 0.5 * m * v²
The mass (m) cancels out in the equation, which simplifies to:
g * h = 0.5 * v²
Rearranging the equation to solve for h, the height of the slide:
h = (0.5 * v²) / g
Plugging in the given values, we have:
h = (0.5 * (7.0 m/s)²) / 9.8 m/s²
By evaluating this equation, we find that the vertical height of the slide is approximately 2.54 meters.