Consider a graph of instantaneous power versus time, with the vertical P-axis st

Question

Consider a graph of instantaneous power versus time, with the vertical P-axis starting at P = 0. What is the physical significance of the area under the P-versus-t curve between vertical lines at t_1 and t_2? A person drops a ball from the roof of a building while a second person observes the motion of the ball from the ground. Both people are stationary. Will the people agree on the value of the gravitational potential energy? The change in gravitational potential energy? The kinetic energy? Explain your answers. When a certain force is applied to an ideal spring, the spring stretches a distance x from its unstretched length and does work W. If instead twice the force is applied, what distance (in terms of x) does the spring stretch from its unstretched length, and how much work (in terms of W) is required to stretch it this distance? A stone of mass m is thrown straight up into the air with speed vo. While in flight, it feels a force of air resistance of magnitude fair. Determine the speed of the stone when it hits the ground. Your answer will contain fair, m, g and vo. You will need to write an equation using either the work energy theorem or the conservation of mechanical energy for the upward path and for the downward path. You will then need to combine the two equations to get the answer. Additionally, for both the upward and downward paths, provide a sentence or two which describe the energy transformations which occur in the system. Use the work-energy theorem to solve each of these problems. You can use Newton's laws to check your answers. Neglect air resistance in all cases. a. A branch falls from the top of a 85.0m tall redwood tree, starting from rest. How fast is it moving when it reaches the ground? b. A volcano ejects a boulder directly upward 625m into the air. How fast was the boulder moving just as it left the volcano? c. A skier moving a 6.00m/s encounters a long, rough horizontal patch of snow having coefficient of kinetic friction 0.220 with her skis. How far does she travel on this patch before stopping? d. Suppose the rough patch in part (c) was only 3.00m long? How fast would the skier be moving when she reached the end of the patch? e. At the base of a frictionless icy hill that rises at 35.0 degree above the horizontal, a sled has a speed of 13.0m/s toward the hill. How high vertically above the base will it go before stopping?

Explanation / Answer

5A.

Vi = 0

a = g

s = 85 m

Using the work-energy theoram

W = dKE

m*g*d = 0.5*m*vf^2 - 0.5*m*vi^2

cancelling m

9.81*85 = 0.5*vf^2

vf = sqrt (2*9.81*85) = 40.84 m/sec

5B.

At max height Vf = 0

a = -g

Vi = ?

Using the equation

m*a*d = 0.5*m*vf^2 - 0.5*m*vi^2

Vi = sqrt (Vf^2 - 2*a*d)

Vi = sqrt (0^2 - 2*(-9.81)*625) = 110.74 m/sec

C.

Vi = 6 m/sec

Ff = -uk*m*g

Vf = 0

s = ?

Using work energy theoram

W = dKE

Ff*d = 0.5*m*vf^2 - 0.5*m*vi^2

(-uk*N)*d = (-uk*m*g)*d = 0.5*m*vf^2 - 0.5*m*vi^2

Since vf = 0

uk*m*g*d = 0.5*m*vi^2

d = 0.5*vi^2/(uk*g)

d = 0.5*6^2/(0.22*9.81) = 8.34 m

5D.

At the distance of 3 m

W = dKE

uk*m*g*d = 0.5*m*vi^2 - 0.5*m*vf^2

vf = sqrt (vi^2 - 2*uk*g*d)

vf = sqrt (6^2 - 2*0.22*9.81*3)

vf = 4.8 m/sec

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