An archer shoots an arrow with a velocity of 45.0m/s at an angle of 50.0degrees with the horizontal.An assistant standing on the level ground 150m downrange from the launch point throws an apple straight up with the minimum intial speed necessary to meet the path of the arrow.(a) what is the initial speed of the apple? (b) at what time after the arrow launch should the apple be thrown so that the arrow hits the apple?
An arrow is shot from an archer’s bow at an angle of 45° with a velocity of 30 m/s. It hits a bullseye on a target after 2 seconds. At what vertical and horizontal displacement is the arrow after that time?
Stwpid
(a) Well, it seems like the apple wants to be a part of the action, trying to meet up with the arrow. What a fruity friend! To figure out the initial speed of the apple, we can use the principle of "hang time". Since the apple and the arrow will take the same amount of time to reach the same vertical position, we can equate their displacement equations. The displacement of the arrow is given by:
d_arrow = v_arrow * t * sin(angle)
And the displacement of the apple will be:
d_apple = v_apple * t + 0.5 * g * t^2
Since they meet up horizontally, the displacement in the horizontal direction will be the same, which means:
d_arrow = d_apple
Plugging in the values we know, we have:
45.0 * t * sin(50.0) = v_apple * t + 0.5 * 9.8 * t^2
Simplifying that equation will give you the initial speed of the apple. But hey, I'm just a clown bot, not a math genius!
(b) As for the second part of your question, we need to find the time at which the apple should be thrown so the arrow hits it. Since we already know the initial speed of the arrow and the angle at which it was launched, we can calculate the time it takes for the arrow to reach the apple's location. Given the vertical position of the apple, we can set up the equation:
d_arrow = v_arrow * t_apple * sin(angle)
Solving for t_apple will give you the time at which the arrow reaches the apple's position. So, cue the timing and precision, and let the apple and arrow meet in mid-air!
Good luck with your calculations! Remember, math can be a real "apple-solute" delight!
To solve this problem, we can break it down into two parts:
Part 1: Finding the initial speed of the apple
In order to find the initial speed of the apple, we need to take into account its vertical and horizontal motion. The vertical motion of the apple can be described by the equation:
y = (v0y * t) - (1/2 * g * t^2)
where:
- y is the vertical displacement (150m)
- v0y is the initial vertical velocity of the apple
- t is the time
- g is the acceleration due to gravity (-9.8m/s^2)
Since the apple is thrown straight up, the initial vertical velocity (v0y) is positive. We also know that the initial horizontal velocity of the apple is zero (since it is thrown straight up), so there is no horizontal acceleration.
Next, we can look at the horizontal motion of the arrow. The horizontal motion of the arrow is given by:
x = v0x * t
where:
- x is the horizontal distance traveled by the arrow (150m)
- v0x is the initial horizontal velocity of the arrow
- t is the time
Since we know the initial velocity of the arrow (45.0m/s) and the launch angle (50.0 degrees with the horizontal), we can find v0x using trigonometry:
v0x = v * cos(theta)
where:
- v is the initial velocity of the arrow (45.0m/s)
- theta is the launch angle (50.0 degrees)
Now that we have both the equations for the vertical motion of the apple and the horizontal motion of the arrow, we need to find the time when the apple meets the path of the arrow. This happens when the horizontal distance traveled by the arrow (x) is equal to the horizontal distance of the apple (150m).
Setting x equal to 150m and solving for t, we get:
150m = v0x * t
Solving this equation will give us the time it takes for the arrow to travel 150m.
Part 2: Finding the time after arrow launch to throw the apple
Once we have the time it takes for the arrow to reach the apple, we can find the time after the arrow launch to throw the apple. This can be calculated by subtracting the time it takes for the arrow to travel 150m from the total time it takes for the arrow to reach the height of the apple.
Let's go ahead and calculate both parts:
Part 1: Finding the initial speed of the apple
1. Calculate v0x:
v0x = v * cos(theta)
v0x = 45.0m/s * cos(50.0 degrees)
v0x = 28.836m/s (rounded to three decimal places)
2. Use the horizontal motion equation to find the time it takes for the arrow to travel 150m:
x = v0x * t
150m = 28.836m/s * t
t = 150m / 28.836m/s
t = 5.204 seconds (rounded to three decimal places)
3. Use the vertical motion equation with the given displacement to find the initial vertical velocity of the apple:
y = (v0y * t) - (1/2 * g * t^2)
150m = (v0y * 5.204s) - (1/2 * 9.8m/s^2 * (5.204s)^2)
150m + 1/2 * 9.8m/s^2 * (5.204s)^2 = v0y * 5.204s
v0y = (150m + 1/2 * 9.8m/s^2 * (5.204s)^2) / 5.204s
v0y = 38.607m/s (rounded to three decimal places)
Therefore, the initial speed of the apple is 38.607m/s (rounded to three decimal places) in the upward direction.
Part 2: Finding the time after arrow launch to throw the apple
1. Subtract the time it takes for the arrow to travel 150m from the total time it takes for the arrow to reach the height of the apple:
Time after arrow launch = Total time - Time to travel 150m
Time after arrow launch = 5.204s - 5.204s
Time after arrow launch = 0s
Therefore, the apple should be thrown immediately after the arrow is launched for the arrow to hit the apple.
Vo = 45m/s @ 50o
Xo = 45*cos50 = 28.93 m/s.
Yo = 45*sin50 = 34.47 m/s.
a. Y^2 = Yo^2 + 2g*h.
h = (Y^2-Yo^2)/2g.
h = (0-(34.47)^2)/-19.6=60.6 m.=Max ht.
Y = Yo + g*t.
Tr = (Y-Yo)/g = (0-34.47)/-9.8=3.52 s.=
Rise time or time to reach max ht.
Xo*T = 150m
28.93*T = 150
T = 5.18 s. = Time in air.
Tf = 5.18 - 3.52 = 1.66 s.
h = hmax - 0.5g*t^2
h = 60.6 - 4.9*(1.66)^2=60.6-13.5=47.1
m above launching level.
d = Vo*t + 0.5g*t^2 = 47.1 m.
Vo*5.18 - 4.9*(5.18)^2 = 47.1
Vo*5.18 - 131.48 = 47.1
Vo*5.18 = 178.58
Vo = 34.47 m/s. = Initial velocity of the apple.
b. T = 0. Apple must be thrown at same time arrow is launched.
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