Concept explainers
A child of mass m starts from rest and slides without friction from a height h along a curved waterslide (Fig. P5.46). She is launched from a height h/5 into the pool. (a) Is mechanical energy conserved? Why? (b) Give the gravitational potential energy associated with the child and her kinetic energy in terms of mgh at the following positions: the top of the waterslide, the launching point, and the point where she lands in the pool. (c) Determine her initial speed V0 at the launch point in terms of g and h. (d) Determine her maximum airborne height ymax in terms of h, g, and the horizontal speed at that height, v0x. (e) Use the x-component of the answer to part (c) to eliminate from the answer to part (d), giving the height ymax in terms of g, h, and the launch angle θ. (f) Would your answers be the same if the waterslide were not frictionless? Explain.
Figure P5.46
(a)
Answer to Problem 46P
Explanation of Solution
Yes. The mechanical energy id conserved in this situation. The forces acting on the child when she slides over the frictionless water slide are the gravitational force and the normal force.
The gravitational force is a conservative force. The normal force is a non-conservative force but it is acting perpendicular to the motion of the child. Therefore the work done due to normal force is zero. The waterslide is frictional less. Thus, the non-conservative force due to friction is also zero.
Thus, the motion of the child in the waterslide is in the absence of the work done by any non-conservative forces. So, the mechanical energy is conserved.
Conclusion:
Yes, the mechanical energy is conserved in this situation.
(b)
Answer to Problem 46P
The kinetic energy of the child at the top of the waterslide is
The kinetic energy of the child at the top of the waterslide is
Explanation of Solution
Section 1:
To determine: The gravitational potential energy and the kinetic energy of the child at the top of the waterslide.
Answer: the kinetic energy of the child at the top of the waterslide is zero and the potential energy is
Explanation:
Given Info:
The height of the waterslide is
The mass of the child is m.
Since the child is at rest in the top of the waterslide, the kinetic energy of the child in the top of the waterslide is zero.
The potential energy of the child at the top of the waterslide is,
- m is the mass of the child
- g is acceleration due to gravity
- h is the height of the waterslide
Therefore the total energy of the child is,
Thus, the kinetic energy of the child at the top of the waterslide is zero and the potential energy is
Section 2:
To determine: The gravitational potential energy and the kinetic energy of the child at the launching point.
Answer: The kinetic energy of the child at the top of the waterslide is
Explanation:
Given Info:
The height of the launching point is
The mass of the child is m.
Formula to calculate the potential energy of the child at the launching point is,
- m is the mass of the child
- g is acceleration due to gravity
- h is the height of the launching point
Substitute
The kinetic energy of the child is,
Thus, the kinetic energy of the child at the top of the waterslide is
Section 3:
To determine: The gravitational potential energy and the kinetic energy of the child at the point where she lands on the pool.
Answer: The kinetic energy of the child at the top of the waterslide is
Explanation:
Given Info:
The height of the pool point is
The mass of the child is m.
Formula to calculate the potential energy of the child at the launching point is,
- m is the mass of the child
- g is acceleration due to gravity
- h is the height of the launching point
Substitute
The kinetic energy of the child is,
Thus, the kinetic energy of the child at the top of the waterslide is
Conclusion:
The kinetic energy of the child at the top of the waterslide is zero and the potential energy is
The kinetic energy of the child at the top of the waterslide is
The kinetic energy of the child at the top of the waterslide is
(c)
Answer to Problem 46P
Explanation of Solution
Given Info:
The kinetic energy of the child at the launching point is
The kinetic energy of the child at the launching point is,
On re-arranging,
The initial speed of the child at the launching point is,
Conclusion:
The initial speed of the child at the launching point is,
(d)
Answer to Problem 46P
Explanation of Solution
When the child leaves the slide, the motion of the child will become a projectile motion. So,
The according to the energy conservation,
Therefore,
The
At the maximum height,
Therefore,
On re-arranging, the maximum airborne height is,
Conclusion:
The maximum airborne height is,
(e)
Answer to Problem 46P
Explanation of Solution
When the launch angle of the child is
On substituting the above result in the result of part d and on use of the part c result,
Conclusion:
The maximum airborne height in terms of h, g and launch angle
(f)
Answer to Problem 46P
Explanation of Solution
No. If the friction is present in the water slide, the above answers will not be the same. When the non-conservative frictional force is present, the mechanical energy will not be conserved.
Thus, when the child moves from the top to the bottom the kinetic energy will be reduced when compared with the case of frictionless waterslide. Therefore, on account of this; the launch speed, maximum height that she reached and the final speed will reduce.
Conclusion:
No. If the friction is present in the water slide, the above answers will not be the same.
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