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You push a box with a force given by the function . Assuming there's no friction forces, and the box is initially at rest at , how fast is the box moving after it has traveled a distance of ?
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Calculating Work by Integration (Calculus)
Scelta multipla
A metal can is pushed from to with a force given by the function . How much work is done by the force on the metal can?
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Guida verificata passo dopo passo1
Identify the physical quantity to find: The problem asks for the work done by a variable force on the metal can as it moves from an initial position to a final position.
Recall the formula for work done by a variable force along a straight line: \(W = \int_{x_i}^{x_f} F(x) \, dx\), where \(x_i\) and \(x_f\) are the initial and final positions, respectively.
Substitute the given force function and limits into the integral: \(W = \int_0^{0.5} 2 e^{-2x} \, dx\).
Set up the integral for evaluation: Recognize that the integral of \(e^{ax}\) is \(\frac{1}{a} e^{ax}\), so you will integrate \(2 e^{-2x}\) accordingly.
Evaluate the definite integral by calculating the antiderivative at the upper and lower limits and subtracting: \(W = \left[ -e^{-2x} \right]_0^{0.5}\).
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