Question

EXAMPLE 7.9 Maximum height of a home-run ball imum height of a projectile launched with ORO Earlier in ample 3.5) we derived an exression for de m initial speed the initial angle ignoring air resistance Video Tutor Solut 2 Now let's derive this expression using a conservation of energy analysis. Final point Velve- SOLUTION SET UP There are no sonconservative forces the work done by gravity is accounted for in the potential energy terms. As shown in Figure 7.30, we choose the launch point as the initial point and the highest point on the trajectory as the final point. We place y at the launch level, then 4 0 and U mh, where his unknown. We can express the kinetic energy at each point in terms of the components of velocity, using the fact that, at each point, u? - + 0, - ( +). Ky = m( ?+06, ?). Initial point o A FIGURE 7.30 Vbe But Do sin , is just they component of initial velocity, which is equal to Making this substitution and solving for h, we find that Dð sin SOLVE Conservation of energy gives K; + - + Uso m( +) + 0 - mu? +0,2) + mgh. To simplify this equation, we multiply through by 2/n to obtain + = + 0,2 + 2gle. Now we recall that, in projectile motion, the component of acceleration is rero, so the x component of velocity is constant, and Also because the final point is the highest point, the vertical component of velocity is zero at that point: -0. Subtracting the terms from both sides, we get V = 2gh. REFLECT This expression is the same as the one we obtained using the kinematic equations for constant acceleration in Example 3.5). The energy approach has the advantage that we don't have to solve for the time when the maximum height is reached not giving the position and/or velocity as an explicit 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%

Answer 1

using conservation of energy

initial kinetic energy = final potential energy

               

mv = mghmar

and h =

võsin(e) 29

therefore , for a given initial speed, if an angle
006 = Ꮎ
,  
h = hmar