a stone is thrown in such a manner that it would just hit a bird at the top of a tree and afterwards reach a maximum height double that of the tree . if at the moment of throwing the stone the bird flies away horizontally with constant velocity and the stone hits the bird after some time . the ratio of horizontal velocity of stone to that of the bird is 1¡n+1¡¡Ìn?
To find the ratio of the horizontal velocity of the stone to that of the bird, let's break down the problem and solve it step by step.
Given information:
- The stone is thrown such that it would hit a bird at the top of the tree.
- After hitting the bird, the stone reaches a maximum height double that of the tree.
- The bird flies away horizontally with a constant velocity.
- We need to find the ratio of the horizontal velocity of the stone to that of the bird.
Let's analyze the motion of the stone and the bird separately.
1. Motion of the Stone:
When the stone is thrown, it follows a parabolic trajectory due to gravity. It reaches its maximum height when its vertical velocity becomes zero. Since the maximum height is double that of the tree, let's assume the height of the tree is "h." So, the maximum height reached by the stone is "2h."
2. Motion of the Bird:
The bird flies away horizontally with a constant velocity. This means the bird has no vertical acceleration, and its vertical velocity remains unchanged. Therefore, the horizontal component of the bird's velocity does not affect the stone's motion.
Now, to determine the ratio of the horizontal velocity of the stone to that of the bird, we need to consider the stone's horizontal velocity and the bird's horizontal velocity.
Let's assume:
- The initial horizontal velocity of the stone is "Vx" (which is the same as the horizontal component of the bird's velocity).
- The time taken for the stone to hit the bird is "t."
Since the stone and the bird both move horizontally, we can use the equation: distance = velocity * time.
For the stone, the horizontal distance covered is equal to the horizontal velocity of the stone multiplied by the time taken for it to hit the bird: (Vx * t).
For the bird, the horizontal distance covered is equal to the horizontal velocity of the bird multiplied by the same time: (Vx * t).
Since the distances are the same, we have:
Vx * t = Vx * t
Now, we can cancel out "Vx" on both sides of the equation, since it is common to both terms:
t = t
This implies that the horizontal velocity of the stone and the bird are the same. Therefore, the ratio of the horizontal velocity of the stone to that of the bird is:
Horizontal Velocity of Stone / Horizontal Velocity of Bird = 1/1 = 1
Hence, the ratio is 1.
To solve this problem, let's break it down into steps:
Step 1: Determine the time it takes for the stone to reach the top of the tree.
Since the maximum height of the stone is double that of the tree, we know that when it reaches its maximum height, the stone will be at the same height as the bird. Let's denote this height as h.
The time it takes for the stone to reach the top of the tree can be determined using the equation for vertical motion: h = (1/2) * g * t^2, where g is the acceleration due to gravity and t is the time.
Since the stone and the bird start at the same height, and the stone reaches its maximum height, we can use this equation to solve for t.
Step 2: Determine the horizontal distance traveled by the bird during the stone's ascent.
Since the bird has a constant horizontal velocity, the distance it travels horizontally during the time it takes for the stone to reach the top of the tree is equal to the horizontal velocity of the bird multiplied by the time. Let's denote this distance as x.
Step 3: Determine the horizontal distance between the stone and the bird at the moment the stone hits the bird.
Since the stone hits the bird after some time, we need to determine the distance between them at that moment. This distance is equal to the horizontal velocity of the bird multiplied by the time it takes for the stone to hit the bird. Let's denote this distance as y.
Step 4: Express the ratio of the horizontal velocity of the stone to that of the bird.
The ratio of the horizontal velocity of the stone to that of the bird is given by the formula: Ratio = (horizontal velocity of the stone) / (horizontal velocity of the bird).
Since the horizontal velocity of the stone is constant throughout its motion, we can express it as the horizontal distance y divided by the time it takes for the stone to hit the bird.
On the other hand, the horizontal velocity of the bird is given by the formula: horizontal velocity of the bird = (horizontal distance x) / (time taken for the stone to reach the top of the tree).
Let's denote the horizontal velocity of the stone as Vstone, the horizontal velocity of the bird as Vbird, and the time it takes for the stone to hit the bird as t2.
Now, let's put all the steps together and calculate the ratio of the horizontal velocity of the stone to that of the bird:
Step 1: Determine the time it takes for the stone to reach the top of the tree:
h = (1/2) * g * t^2
2h = g * t^2
t = sqrt((2h)/g)
Step 2: Determine the horizontal distance traveled by the bird during the stone's ascent:
x = Vbird * t
Step 3: Determine the horizontal distance between the stone and the bird at the moment the stone hits the bird:
y = Vstone * t2
Step 4: Express the ratio of the horizontal velocity of the stone to that of the bird:
Ratio = (Vstone) / (Vbird)
Ratio = (y / t2) / (x / t)
By substituting the values from steps 1, 2, and 3 into step 4, we can calculate the ratio.
Well, to make it easy, let's say the tree is 50 meters high. Then the max stone height will be 100 meters.
That gives us a vertical problem to solve.
What is Vi, initial vertical velocity?
(1/2) m Vi^2 = m g h
so
Vi = sqrt(2 g h = sqrt (200*9.81) = 44.3 m/s
so when will it get to the tree top?
h = Vi t - 4.9 t^2
50 = 44.3 t - 4.9 t^2
4.9 t^2 -44 t + 50 = 0
t = [ 44 +/- sqrt(1936 -980) ] /9.81
t = [ 44 +/- 30.9 ] /9.81
t = 1.34 seconds on the way up, WHERE IT WOULD HAVE HIT THE BIRD
t = 7.64 seconds on the way down - WHEN IT HITS THE BIRD!
the bird flew for 7.64 seconds at speed u so went 7.64 u meters from the tree
The stone went at horizontal speed s so went a total of 7.64 s meters
of which 1.34 s was the distance from the thrower to the tree. so (7.64-1.34) s = 7.64 u
I guess you can find s/u now :)