This question aims to find the height of the sprinter where the gravitational potential energy is equal to kinetic energy for the world’s fastest human who can reach the speed of 11m/s. The kinetic energy of an object is due to its motion. When work is done on an object by applying a net force that transfers energy, the object accelerates, thereby gaining kinetic energy.
Kinetic energy is given by the formula:
\[K=\dfrac{1}{2}mv^2\]
The potential of the potential object arises from this position. For example, a heavy ball in a demolition machine stores energy when it is high. This stored potential is called potential energy. Depending on the position, the taut bow can also conserve energy. Gravity or gravitational force can be a huge object in relation to something larger because of the force of gravity. The potential energy associated with the field of gravity is released (converted into kinetic energy) as objects cross each other.
Gravitational potential energy is given by the formula:
\[U=mgh\]
Expert Answer
Speed is given in the question as:
\[v_{human}=v=11\dfrac{m}{s}\]
Gravitational potential energy is given as:
\[U=mgh\]
kinetic energy is given as:
\[K=\dfrac{1}{2}mv^2\]
$g$ is given as gravitational acceleration constant and its value is given as:
\[g=9.8\dfrac{m}{s^2}\]
To increase the gravitational potential energy by an amount equal to the kinetic energy at full speed, the kinetic energy must be equal to the gravitational potential energy.
\[K=U\]
\[\dfrac{1}{2}mv^2=mgh\]
\[\dfrac{v^2}{2}=gh\]
\[h=\dfrac{v^2}{2g}\]
Plug the values of the gravity $g$ and speed $v$ into the formula to calculate height.
\[h=\dfrac{11^2}{2\times9.8}\]
\[h=6.17m\]
He needs to climb $6.17m$ above the ground.
Numerical Result
The person needs to climb $6.17m$ above the ground in order to make kinetic energy equal to gravitational potential energy.
Example
The world’s fastest humans can reach speeds of about $20\dfrac{m}{s}$. How high does such a sprinter have to climb to increase the gravitational potential energy by an amount equal to the kinetic energy at full speed?
Speed is given as:
\[v_{human}=v=20\dfrac{m}{s}\]
Gravitational potential energy is given as:
\[U=mgh\]
kinetic energy is given as:
\[K=\dfrac{1}{2}mv^2\]
“g” is given as gravitational acceleration constant and its value is given as:
\[g=9.8\dfrac{m}{s^2}\]
To increase the gravitational potential energy by an amount equal to the kinetic energy at full speed, the kinetic energy must be equal to the gravitational potential energy.
\[K=U\]
\[\dfrac{1}{2}mv^2=mgh\]
\[\dfrac{v^2}{2}=gh\]
\[h=\dfrac{v^2}{2g}\]
Plug the values of the gravity $g$ and speed $v$ into the formula to calculate height.
\[h=\dfrac{20^2}{2\times9.8}\]
\[h=20.4m\]
He needs to climb $20.4m$ above the ground.
The person needs to climb $20.4m$ above the ground in order to make kinetic energy equal to gravitational potential energy.