Therefore, the speed of the raft is 58. REASONING The stone that is thrown upward loses speed on the way up. The stone that is thrown downward gains speed on the way down. The stones cross paths below the point that corresponds to half the height of the cliff. To see why, consider where they would cross paths if they each maintained their initial speed as they moved. Then, they would cross paths exactly at the halfway point. However, the stone traveling upward begins immediately to lose speed, while the stone traveling downward immediately gains speed. Thus, the upward moving stone travels more slowly than the downward moving stone. Consequently, the stone thrown downward has traveled farther when it reaches the crossing point than the stone thrown upward. The initial velocity is known for both stones, as is the acceleration due to gravity. In v0 a addition, we know that at the crossing point the stones are at the same place at the same time Furthermore, the position of each stone is specified by its displacement from its t. y starting point. The equation of kinematics that relates the variables and is v0, a, t y Equation 2.8 , and we will use it in our solution. In using this equation, we will assume upward to be the positive direction. Applying Equation 2.8 to each stone, we have SOLUTION In these expressions is the time it takes for either stone to t reach the crossing point, and is the acceleration due to a gravity. Note that is the displacement of the upward yup moving stone above the base of the cliff, is the ydown displacement of the downward moving stone below the top of the cliff, and is the displacement of the cliff-top H above the base of the cliff, as the drawing shows. The distances above and below the crossing point must add to equal the height of the cliff, so we have the minus sign appears because the displacement points in the negative direction. ydown Substituting the two expressions for and into this equation gives yup ydown This equation can be solved for to show that the travel time to the crossing point is t
Therefore, the speed of the raft is 58. REASONING The stone that is thrown upward loses speed on the way up. The stone that is thrown downward gains speed on the way down. The stones cross paths below the point that corresponds to half the height of the cliff. To see why, consider where they would cross paths if they each maintained their initial speed as they moved. Then, they would cross paths exactly at the halfway point. However, the stone traveling upward begins immediately to lose speed, while the stone traveling downward immediately gains speed. Thus, the upward moving stone travels more slowly than the downward moving stone. Consequently, the stone thrown downward has traveled farther when it reaches the crossing point than the stone thrown upward. The initial velocity v0 is known for both stones, as is the acceleration a due to gravity. In addition, we know that at the crossing point the stones are at the same place at the same time t. Furthermore, the position of each stone is specified by its displacement y from its starting point. The equation of kinematics that relates the variables v0, a, t and y is Equation 2.8 , and we will use it in our solution. In using this equation, we will assume upward to be the positive direction. SOLUTION Applying Equation 2.8 to each stone, we have In these expressions t is the time it takes for either stone to reach the crossing point, and a is the acceleration due to gravity. Note that yup is the displacement of the upward moving stone above the base of the cliff, ydown is the displacement of the downward moving stone below the top of the cliff, and H is the displacement of the cliff-top above the base of the cliff, as the drawing shows. The distances above and below the crossing point must add to equal the height of the cliff, so we have the minus sign appears because the displacement ydown points in the negative direction. Substituting the two expressions for yup and ydown into this equation gives This equation can be solved for t to show that the travel time to the crossing point is