Question

An object of mass \(40\) \(kg\) is raised to a height of \(5\) \(m\) above the ground. What is its potential energy? If the object is allowed to fall, find its kinetic energy when it is half-way down.

Answer 1

Step 1: Calculate the Potential Energy (PE) at the height of 5 meters
Potential energy is given by the formula:
\(\text{PE} = mgh\)
where:
m is the mass of the object (40 kg),
g is the acceleration due to gravity (approximately 9.8 m/s²),
h is the height above the ground (5 m).
\(\text{PE} = 40 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 5 \, \text{m}\)
\(\text{PE} = 40 \times 9.8 \times 5\)
\(\text{PE} = 1960 \, \text{J}\)
So, the potential energy at 5 meters is 1960 joules.

Step 2: Calculate the Kinetic Energy (KE) when the object is halfway down
When the object is halfway down, it has fallen a distance of 2.5 meters (since 5m/2 = 2.5 m). At this point, its height above the ground is 2.5 meters.
The potential energy at this new height is:
\(\text{PE}_{\text{new}} = mgh_{\text{new}}\)
where \(h_{\text{new}} = 2.5 \, \text{m}\).

\(\text{PE}_{\text{new}} = 40 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 2.5 \, \text{m}\)
\(\text{PE}_{\text{new}} = 40 \times 9.8 \times 2.5\)
\(\text{PE}_{\text{new}} = 980 \, \text{J}\)

Since energy is conserved, the loss in potential energy equals the gain in kinetic energy as the object falls. Initially, the potential energy was 1960 joules. At half the height, the potential energy is 980 joules. The difference is converted to kinetic energy.

\(\text{KE} = \text{Initial PE} - \text{PE}_{\text{new}}\)
\(\text{KE} = 1960 \, \text{J} - 980 \, \text{J}\)
\(\text{KE} = 980 \, \text{J}\)

So, the answer is 980J.