Question 11.SP.12: A conveyor belt at an angle of 20º with the horizontal is us...
A conveyor belt at an angle of 20º with the horizontal is used to trans-fer small packages to other parts of an industrial plant. A worker tosses a package with an initial velocity v0 at an angle of 45º so that its velocity is parallel to the belt as it lands 1 m above the release point. Determine (a) the magnitude of v_{0}, (b) the horizontal distance d.
STRATEGY: This is a projectile motion problem, so you can con-sider the vertical and the horizontal motions separately. First deter-mine the equations governing the motion in each direction, then use them to determine the unknown quantities.
MODELING and ANALYSIS:
Horizontal Motion. Placing the axes of your origin at the location where the package leaves the workers hands (Fig. 1), you can write
Landing on the Belt. The problem statement indicates that when the package lands on the belt, its velocity vector will be in the same direction as the belt is moving. If this happens when t = t_{1}, you can write
\frac{v_{y}}{v_{x}}=\tan 20^{\circ}=\frac{v_{0} \sin 45^{\circ}-g t_{1}}{v_{0} \cos 45^{\circ}}=1-\frac{g t_{1}}{v_{0} \cos 45^{\circ}} (1)This equation has two unknown quantities: t_{1} and v_{0}. Therefore, you need more equations. Substituting t = t_1 into the remaining projectile motion equations gives
\begin{aligned} d &=\left(v_{0} \cos 45^{\circ}\right) t (2)\\1 \mathrm{~m} &=\left(v_{0} \sin 45^{\circ}\right) t_{1}-\frac{1}{2} g t_{1}^{2} (3)\end{aligned}You now have three equations (1), (2), and (3) and three unknowns t_{1}, v_{0}, and d. Using g = 9.81 m/s² and solving these three equations give t_{1} = 0.3083 s and
\begin{array}{r}v_{0}=6.73 \mathrm{~m} / \mathrm{s} \\d=1.466 \mathrm{~m}\end{array}REFLECT and THINK: All of these projectile problems are simi-lar. You write down the governing equations for motion in the hori-zontal and vertical directions and then use additional information in the problem statement to solve the problem. In this case, the distance is just less than 1.5 meters, which is a reasonable distance for a worker to toss a package.
