A spring-loaded toy gun exerts a variable force on a plastic ball as the spring expands. Consider a horizontal spring and a ball of mass m whose position when barely touching a fully expanded spring is . The ball is pushed to the left, compressing the spring. You’ll learn in Chapter that the spring force on the ball, when the ball is at position (which is negative), can be written as , where k is called the spring constant. The minus sign is needed to make the -component of the force positive. Suppose the ball is initially pushed to , then released and shot to the right.
a. Use what you’ve learned in calculus to prove that
b. Find an expression, in terms of , , and , for the speed of the ball as it comes off the spring at .
(a) The calculation is proved
(b) Expression for the speed of the ball as it comes of the spring is ,
Calculus is that the study of change in mathematics, the same as how geometry is that the analysis of shape while algebra is that the research of operations and the way they're applied to unravel problems.
The expression of ball velocity when spring leaves,
Expand equations using values,
Integrate the equations,
Very small objects, such as dust particles, experience a linear drag force, (bv, direction opposite the motion), where is a constant. That is, the quadratic model of drag of Equation fails for very small particles. For a sphere of radius , the drag constant can be shown to be , where is the viscosity of the gas.
a. Find an expression for the terminal speed of a spherical particle of radius and mass falling through a gas of viscosity .
b. Suppose a gust of wind has carried a -diameter dust particle to a height of . If the wind suddenly stops, how long will it take the dust particle to settle back to the ground? Dust has a density of the viscosity of air is , and you can assume that the falling dust particle reaches terminal speed almost instantly.
Given free body diagram :
a. Write a realistic dynamics problem for which this is the correct free-body diagram. Your problem should ask a question that can be answered with a value of position or velocity (such as “How far?” or “How fast?”), and should give sufficient information to allow a solution.
b. Solve your problem!
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