is zero, so the x component of velocity is constant, and Also. energy approach h
ID: 1642692 • Letter: I
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is zero, so the x component of velocity is constant, and Also. energy approach has the advantage that we don't have to solve for because the final point is the highest point, the vertical component of time when the maximum height is reached. It has the disadvantage velocity is zero at that point: 0. Subtracting the u terms from not giving the position and/or velocity as an explicit function ofu both sides, we get Iol,Me JuOmponent of acceleration the hessio ls the Sanme as the one we oblaincd usig in Example 3.5). The energy approach has the advantage that we don't have to solve for the of kinematic equations for constant acceleration (in Example) Ihe function of time. Practice Problem: Use a conservation-of-energy analysis to show that. for a given initial speed, the maximum height is greatest when = 90° EXAMPLE 7.10 Calculating speed along a vertical circle Here we will tackle a circular-motion problem using conservation of energy. Because the acceleration in this problem is not constant, we cannot approach it with the tools we used when we studied circular mo. tion in Chapter 4. A boy skateboards down a quarter-pipe with radius R - 3.0 m (Figure 7.31). The total mass of the boy and the skateboard is 25.0 kg. If he starts from rest and there is no friction, derive an al- gebraic expression for his speed at the bottom of the ramp. Evaluate this expression with the values given The skateboarder's center of mass moves in a circle with radius somewhat smaller than R: ignore this small difference.) Video Tutor Solution Initial point Initial point0 ui = 0 m 25.0 kg s small m2 R930m At each point. the normal force acts perpendicular to the direction of displacement, so only his weight does work on him. Final point Final point FIGURE 7.31 (a) Motion showing initial and final points. (b) Free-body diagrams for initial and final points.Explanation / Answer
This is basically a problem of conservation of energy.
Let us consider the datum as the lower point of the curve. The potential energy of the boy will be calculated with respect to this datum.
Velocity of boy at topmost point = 0m/s
Height of boy at highest point with respect to datum = h=3m
Potential energy of boy = mgh = 25*9.8*3 = 735 J
Kinetic energy of boy = 0
Total energy of boy at topmost point = 735+0 = 735 J
Potential energy of boy at lowest point = 0
Let speed at lowest point be v
kinetic energy of boy at lowest point = (1/2)*25*v2
Total energy of boy at lowest point = 0+(1/2)*25*v2 = (1/2)*25*v2 = 12.5v2
From conservation of energy, total energy at highest point = total energy at lowest point
735 = 12.5v2
v = sqrt(735/12.5) = 7.67 m/s
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