To find the initial speed of the motorcycle, we can use the principle of conservation of energy. The initial kinetic energy of the motorcycle will be converted into potential energy as it climbs the hill, and some energy will be lost due to friction.
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Potential Energy at Maximum Height: The potential energy (PE) at the maximum height can be calculated by the formula: \[ PE = mgh \] where \( m \) is the mass (250.0 kg), \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)), and \( h \) is the height (250.0 m).
Substituting the values: \[ PE = 250.0 , \text{kg} \times 9.81 , \text{m/s}^2 \times 250.0 , \text{m} = 250.0 \times 9.81 \times 250.0 \] \[ PE = 250.0 \times 2452.5 = 613125 , \text{J} \]
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Total Energy Initially: The initial kinetic energy (KE) of the motorcycle can be expressed as: \[ KE_{initial} = PE + \text{Energy dissipated by friction} \] From the problem, we have: \[ \text{Energy dissipated by friction} = 2.55 \times 10^5 \text{ J} \] Therefore: \[ KE_{initial} = 613125 , \text{J} + 2.55 \times 10^5 , \text{J} = 613125 , \text{J} + 255000 , \text{J} = 868125 , \text{J} \]
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Initial Kinetic Energy in Terms of Speed: The initial kinetic energy can also be calculated using the formula for kinetic energy: \[ KE = \frac{1}{2} mv^2 \] Setting the two expressions for kinetic energy equal to each other: \[ \frac{1}{2} (250.0 , \text{kg}) v^2 = 868125 , \text{J} \]
Solving for \( v^2 \): \[ 125 v^2 = 868125 \] \[ v^2 = \frac{868125}{125} = 6945 \] \[ v = \sqrt{6945} \approx 83.33 , \text{m/s} \]
Thus, the initial speed of the motorcycle was approximately 83.33 m/s.