A bicycle and rider have a combined potential energy of 1400 J. What is the potential energy if the bicycle and rider have one fourth their mass at the same height?
That's all the question gives.
PE = mgh
so,
(1/4 m)gh = 1/4 mgh = 1/4 PE
To solve this problem, we need to use the equation for potential energy:
Potential Energy (PE) = mass (m) x gravity (g) x height (h)
Given that the bicycle and rider have a combined potential energy of 1400 J, let's assume their mass is represented by 'm' and the height is represented by 'h'.
Therefore, we can express the given situation as:
PE1 = m x g x h
Now, if the bicycle and rider have one fourth their mass at the same height, we can represent the new mass as 'm/4'. Given that potential energy depends on mass, height, and gravity, we can use the equation to represent the situation as:
PE2 = (m/4) x g x h
We are asked to find the potential energy in this new situation.
To find the relationship between PE1 and PE2, we can divide the second equation by the first equation:
PE2/PE1 = ((m/4) x g x h) / (m x g x h)
We can simplify this expression by canceling out the common terms:
PE2/PE1 = 1/4
This tells us that the potential energy in the second situation is one fourth (1/4) of the potential energy in the first situation.
To find the potential energy of the bicycle and rider when their mass is one fourth, we can multiply the initial potential energy by 1/4:
PE2 = PE1 x (1/4)
PE2 = 1400 J x (1/4)
PE2 = 350 J
Therefore, the potential energy of the bicycle and rider when their mass is one fourth is 350 J.
To solve this question, let's break it down step by step.
First, we need to understand the concept of potential energy. Potential energy is the energy an object possesses due to its position or state. In this case, we are considering the potential energy of a bicycle and rider.
The question states that the bicycle and rider have a combined potential energy of 1400 J. Let's denote this potential energy as PE_original.
Now, the question asks what the potential energy would be if the bicycle and rider have one-fourth their mass at the same height. To find the potential energy in this scenario, we can use the principle of conservation of mechanical energy.
According to the principle of conservation of mechanical energy, the total mechanical energy of a system remains constant as long as no external forces or work is done on that system.
In this case, the only form of energy we are considering is potential energy, so we can say that the initial potential energy (PE_original) is equal to the final potential energy (PE_final).
Given that the mass of the bicycle and rider is reduced to one-fourth, the mass is decreased by a factor of 4. Therefore, the final potential energy (PE_final) can be calculated using the equation:
PE_final = (mass_final / mass_original) * PE_original
Since the mass_final is one-fourth the mass_original, we can substitute these values in the equation:
PE_final = (1/4) * PE_original
Let's substitute the given value of PE_original to find the final potential energy:
PE_final = (1/4) * 1400 J
Calculating this expression:
PE_final = 350 J
Hence, the potential energy of the bicycle and rider, when their mass is reduced to one-fourth, is 350 J.