From a tree that is 15m tall, a fruit of mass 2kg falls to the ground. (a) find the initial GPE (b) find the speed at impact (c) find the KE at impact (d) how much energy was lost to heat and sound after the mass came to rest?
To answer these questions, we need to consider the principles of gravitational potential energy (GPE), kinetic energy (KE), and energy conservation.
(a) Finding the initial GPE:
The formula for calculating gravitational potential energy is: GPE = m * g * h, where m is the mass, g is the acceleration due to gravity (around 9.8 m/s² on Earth), and h is the height.
In this case, the mass of the fruit is 2 kg, and the height it fell from is 15 m. So, the initial GPE is: GPE = 2 kg * 9.8 m/s² * 15 m = 294 J.
(b) Finding the speed at impact:
To find the speed at impact, we can use the principle of conservation of energy. The initial GPE is converted to kinetic energy at the point of impact.
The formula for calculating kinetic energy is: KE = (1/2) * m * v², where m is the mass and v is the velocity.
In this case, we know the mass (2 kg) but not the velocity. However, we can calculate the velocity using another equation:
The principle of conservation of energy states that the initial GPE is equal to the final KE. So, 294 J = (1/2) * 2 kg * v².
Rearranging the equation to solve for v, we get: v = √(2 * 294 J / 2 kg) ≈ 12.7 m/s.
Therefore, the speed at impact is approximately 12.7 m/s.
(c) Finding the KE at impact:
Using the derived velocity of 12.7 m/s, we can calculate the kinetic energy at impact using the kinetic energy formula:
KE = (1/2) * 2 kg * (12.7 m/s)² = 161.3 J.
Therefore, the kinetic energy at impact is 161.3 J.
(d) Calculating the energy lost to heat and sound after coming to rest:
The energy lost to heat and sound can be calculated by subtracting the final GPE from the initial GPE.
Since the fruit has come to rest, the final GPE is zero. Therefore, the energy lost to heat and sound is equal to the initial GPE:
Energy lost = Initial GPE = 294 J.
Hence, 294 J of energy was lost to heat and sound after the mass came to rest.