a stone is projected with a speed of 120m/s at an angle of 30 degrees to the horizontal to hit a monkey following from a branch of a tree at 68m high above the ground.assumming both the stone and the monkey started their motions at the sane tone calculate the velocity of the falling monkey at the point it collided with the stone
To solve this problem, we can break it down into horizontal and vertical components.
1. Horizontal Component:
The initial velocity of the stone, V0x, in the horizontal direction is given by:
V0x = V0 * cosθ
where V0 is the initial speed of the stone and θ is the angle with respect to the horizontal.
V0x = 120 m/s * cos(30°)
V0x = 120 m/s * √(3)/2
V0x ≈ 103.92 m/s
2. Vertical Component:
The initial velocity of the stone, V0y, in the vertical direction is given by:
V0y = V0 * sinθ
V0y = 120 m/s * sin(30°)
V0y = 120 m/s * 1/2
V0y = 60 m/s
Now, let's analyze the motion of the monkey. The stone and the monkey started their motions at the same time, so let's find the time it takes for the stone to reach the monkey.
3. Finding Time:
The vertical distance covered by the stone, Δy, is the height of the monkey above the ground.
Δy = 68 m
Using the equation of motion:
Δy = V0y * t + (1/2) * g * t^2
68 = 60 * t + (1/2) * 9.8 * t^2
68 = 60t + 4.9t^2
Rearranging the equation:
4.9t^2 + 60t - 68 = 0
Solving this quadratic equation, we get two solutions: t = -2.45 s or t ≈ 2.83 s.
Since time cannot be negative, we consider t = 2.83 s as the time it takes for the stone to reach the monkey.
4. Velocity of the Monkey:
Now, let's calculate the velocity of the monkey when it is hit by the stone at t = 2.83 s.
Using the equation of motion:
Vf = Vi + gt
Vertical component:
Vfy = V0y + g*t
Vfy = 60 m/s + 9.8 m/s^2 * 2.83 s
Vfy ≈ 88.59 m/s
Therefore, the velocity of the falling monkey at the point it collided with the stone is approximately 88.59 m/s.
a stone is projected with a speed of 120m/s at an angle of 30 degrees to the horizontal to hit a monkey following from a branch of a tree at 68m high above the ground.assumming both the stone and the monkey started their motions at the sane tone calculate the velocity of the falling monkey at the point it collided with the stone
To find the velocity of the falling monkey at the point of collision with the stone, we need to consider the vertical motion of both the stone and the monkey.
1. Vertical Motion of the Stone:
The initial vertical velocity of the stone, V0y, is given by:
V0y = V0 * sinθ
V0y = 120 m/s * sin(30°)
V0y = 60 m/s
We can use the equation of motion to find the time it takes for the stone to reach the height of the monkey.
Δy = V0y * t - (1/2) * g * t^2
68 m = 60 m/s * t - (1/2) * 9.8 m/s^2 * t^2
Rearranging the equation:
4.9 t^2 - 60t + 68 = 0
Solving this quadratic equation, we find two solutions: t = 9.35 s or t ≈ 1.46 s.
Since the time cannot be 9.35 s (as it is too large for the given scenario), we consider t ≈ 1.46 s as the time it takes for the stone to reach the height of the monkey.
2. Vertical Motion of the Monkey:
The monkey falls from rest, so its initial vertical velocity, V0y, is 0 m/s.
Using the equation of motion:
Δy = V0y * t + (1/2) * g * t^2
68 m = 0 * t + (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation:
4.9 t^2 = 68
Solving for t, we get t ≈ 2.97 s.
Now, let's calculate the velocity of the monkey when it is hit by the stone at t = 1.46 s.
3. Velocity of the Monkey:
Using the equation of motion:
Vf = Vi + g*t
Vertical component:
Vfy = V0y + g*t
Vfy = 0 m/s + 9.8 m/s^2 * 1.46 s
Vfy ≈ 14.25 m/s
Therefore, the velocity of the falling monkey at the point of collision with the stone is approximately 14.25 m/s.
To find the velocity of the falling monkey at the point it collides with the stone, we need to use projectile motion equations.
Step 1: Find the time of flight for the stone:
The time of flight for a projectile can be calculated using the formula:
t = (2 * usinθ) / g
where u is the initial velocity, θ is the angle above the horizontal, and g is the acceleration due to gravity.
Given:
u (initial velocity) = 120 m/s
θ (angle) = 30 degrees
g (acceleration due to gravity) = 9.8 m/s²
Let's substitute the values into the formula:
t = (2 * 120 * sin(30)) / 9.8
Step 2: Calculate the horizontal distance traveled by the stone:
The horizontal distance traveled by the stone can be found using the formula:
x = ucosθ * t
Given:
u (initial velocity) = 120 m/s
θ (angle) = 30 degrees
t (time of flight) = calculated in Step 1
Let's substitute the values into the formula:
x = 120 * cos(30) * t
Step 3: Calculate the vertical distance traveled by the stone:
The vertical distance traveled by the stone can be found using the formula:
y = usinθ * t - (1/2)gt²
Given:
u (initial velocity) = 120 m/s
θ (angle) = 30 degrees
t (time of flight) = calculated in Step 1
g (acceleration due to gravity) = 9.8 m/s²
Let's substitute the values into the formula:
y = 120 * sin(30) * t - (1/2) * 9.8 * t²
Step 4: Calculate the vertical velocity of the monkey when it collides with the stone:
Since the stone and the monkey start their motion at the same time, the monkey falls from a height of 68 m. Therefore, the vertical distance traveled by the monkey is equal to 68 m.
Let's equate the vertical distance traveled by the monkey (68 m) with the formula for vertical distance traveled by the stone (from Step 3):
68 = 120 * sin(30) * t - (1/2) * 9.8 * t²
We can solve this equation to find the value of t.
Step 5: Calculate the vertical velocity of the monkey at the point of collision:
To calculate the vertical velocity of the monkey at the point of collision, we need to differentiate the equation from Step 3 with respect to time (t) and solve for the velocity.
dy/dt = u * sinθ - gt
Given:
u (initial velocity) = 120 m/s
θ (angle) = 30 degrees
g (acceleration due to gravity) = 9.8 m/s²
t (time of collision) = calculated in Step 4
Let's substitute the values into the formula:
v = 120 * sin(30) - 9.8 * t
Now, substitute the value of t from Step 4 into the equation to calculate the vertical velocity (v) at the point of collision.