Query 10 Differential Equations

Question:  3.7.4.  Solve the equation m dv/dt = - k v / (1 + x), where x is position as a function of t and v is velocity as a function of t.

How far does the object travel before coming to rest?

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3.7.6.  A vertical projectile has initial velocity v_0, and experiences drag force k v^2 in the direction opposite its motion.  Assume that acceleration g of gravity is constant.  Find the maximum height to which the projectile rises.

If the initial velocity is 80 m/s and the projectile, whose mass is .12 grams, rises to a height of 40 meters (estimated quantities for a plastic BB), what is the value of k?

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3.7.8.  Mass m accelerates from velocity v_1 to velocity v_2, while constant power is exerted by the net force.  At any instant power = force * velocity (physics explanation:  power = dw / dt, the rate at which work is being done; w = F * dx so dw/dt = F * dx/dt = F * v).

How far does the mass travel as it accelerates?

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3.7.11.  An object falls to the surface of the Earth from a great height h, and as it falls experiences a drag force proportional to the square of its velocity.  Assume that the gravitational force is - G M m / r^2.

What will be its impact velocity?

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