The circus has come to town, and Lael the Magnificent will kick a grape into her assistant's waiting mouth (not to worry − the grape is freshly washed). She starts at the origin and boots the grape at an angle 20.0° east of north. The grape starts from a height of 0.200 meters, with a velocity of 12.0 m/s, 30.0° above the horizontal. Her assistant races to intercept the grape as it descends. Assume that his mouth is 1.75 meters above the ground. (a) How much time elapses after the kick until he catches the grape? (b) At what position must he be standing for the grape to fall into his mouth? Use the x dimension for east-west, and the y dimension for north-south.
PLS HELP!!!! I REALLY REALLY NEED SOMEONE TO HELP FIGURE THIS OUT!
To solve this problem, we need to analyze the motion of the grape and the assistant separately. Let's break it down step by step.
(a) To find the time it takes for the assistant to catch the grape, we first need to find the time it takes for the grape to reach the assistant's mouth.
1. Split the initial velocity of the grape into its vertical and horizontal components:
Vertical component (V_y) = velocity * sin(angle) = 12.0 m/s * sin(30.0°)
Horizontal component (V_x) = velocity * cos(angle) = 12.0 m/s * cos(30.0°)
2. Determine the time it takes for the grape to reach its maximum height:
The vertical motion of the grape is influenced by gravity only, so we can use the equation: V_f = V_i + a * t
Since V_f = 0 (at maximum height), V_i = V_y, and a = -9.8 m/s^2 (acceleration due to gravity), we can solve for t:
0 = V_y + (-9.8 m/s^2) * t
t = V_y / 9.8 m/s^2
3. Calculate the time it takes for the grape to hit the ground:
The vertical motion of the grape can be determined by the equation:
y = V_i * t + (1/2) * a * t^2
Here, y = -0.200 m (negative because the displacement is downward), V_i = V_y, and a = -9.8 m/s^2.
Rearranging the equation, we have:
0 = V_y * t + (1/2) * (-9.8 m/s^2) * t^2
Solve this quadratic equation for t using the quadratic formula.
4. Add the times obtained in steps 2 and 3 to get the total time it takes for the grape to hit the ground.
(b) To find the position where the assistant must be standing, we need to calculate the horizontal displacement of the grape.
1. Calculate the horizontal displacement (X_displacement) of the grape using the equation:
X_displacement = V_x * t
Use the time obtained in part (a) to find the horizontal displacement.
2. Determine the position where the assistant must be standing by considering the horizontal displacement from the origin.
With these calculations, you should be able to determine the answers to (a) and (b) of the given problem.