What is the kinetic energy of the boy and the bicycle after rolling 20m down the slope, given the combined mass is 25kg and there is a frictional force of 30N?
Understand the Problem
The question is asking to calculate the kinetic energy of a boy on a bicycle after rolling down a slope, considering the initial potential energy, the work done against friction, and the distance rolled down the slope. Specifically, it combines concepts of gravitational potential energy, kinetic energy, and friction.
Answer
The kinetic energy is approximately $1850 \, \text{J}$.
Answer for screen readers
The kinetic energy of the boy and the bicycle after rolling 20m down the slope is approximately ( 1850 , \text{J} ).
Steps to Solve
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Calculate Initial Potential Energy The potential energy (PE) at the top of the slope can be calculated using the formula: $$ PE = mgh $$ Here, ( m = 25 , \text{kg} ) and ( g = 9.81 , \text{m/s}^2 ). The height ( h ) can be found using the incline angle ( \theta = 30^\circ ) and the distance ( d = 20 , \text{m} ): $$ h = d \sin(\theta) = 20 \sin(30^\circ) = 20 \cdot 0.5 = 10 , \text{m} $$ Now, substituting the values: $$ PE = 25 \cdot 9.81 \cdot 10 = 2452.5 , \text{J} $$
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Calculate Work Done Against Friction The work done against friction (W_f) over the distance rolled can be calculated as: $$ W_f = F_f \cdot d $$ Here, the frictional force ( F_f = 30 , \text{N} ) and ( d = 20 , \text{m} ): $$ W_f = 30 \cdot 20 = 600 , \text{J} $$
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Calculate Kinetic Energy The kinetic energy (KE) after rolling down the slope can be calculated by subtracting the work done against friction from the initial potential energy: $$ KE = PE - W_f $$ Substituting the values: $$ KE = 2452.5 - 600 = 1852.5 , \text{J} $$
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Final Kinetic Energy Approximation Since the answers are in whole numbers, we can round ( 1852.5 , \text{J} ) to ( 1850 , \text{J} ).
The kinetic energy of the boy and the bicycle after rolling 20m down the slope is approximately ( 1850 , \text{J} ).
More Information
The calculation shows how potential energy is transformed into kinetic energy, while accounting for the work done against friction. This illustrates the principle of conservation of energy, modified by the presence of non-conservative forces like friction.
Tips
- Forgetting to convert angles from degrees to radians if using calculators set in radian mode.
- Not accounting for the height calculation separately, which can lead to incorrect potential energy values.
- Miscalculating the work done against friction by omitting the distance.
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