a rock is thrown horizontally off a 100m cliff making an angle of 50.8 degree. it land 95m away .at what speed was it thrown?
To find the speed at which the rock was thrown, we can use the following equation for projectile motion:
d = (v^2 * sin(2θ)) / g
Where:
- d is the horizontal distance traveled (95 m)
- v is the initial velocity of the rock
- θ is the angle of the throw (50.8 degrees)
- g is the acceleration due to gravity (9.8 m/s^2)
Rearranging the equation to solve for v:
v^2 = (d * g) / sin(2θ)
Plugging in the given values:
v^2 = (95 * 9.8) / sin(2 * 50.8)
v^2 = 922.1 / sin(101.6)
Using a calculator:
v^2 ≈ 149.12
Taking the square root of both sides:
v ≈ √149.12
v ≈ 12.2 m/s
Therefore, the rock was thrown at approximately 12.2 m/s.
To find the initial speed at which the rock was thrown, we can use the equation of motion in the horizontal direction:
horizontal distance = initial horizontal velocity × time
Given that the rock lands 95m away and the angle made by the rock with the horizontal is 50.8 degrees, we can break down the initial velocity into horizontal and vertical components.
Let's denote:
- v: initial speed (magnitude of the velocity vector)
- θ: launch angle (50.8 degrees in this case)
- g: acceleration due to gravity (approximately 9.8 m/s^2)
The horizontal component of the initial velocity (v_x) is given by:
v_x = v × cos(θ) --> (1)
The time of flight (t) can be determined from the vertical motion of the rock using the equation:
vertical distance = initial vertical velocity × time + (1/2) × g × time^2
In this case, the rock is dropped from rest, so the initial vertical velocity is zero. Hence, for the vertical motion, we have:
100m = (1/2) × g × t^2 --> (2)
Now, we can solve equation (1) for v:
v = v_x / cos(θ)
To find v_x, we'll use the equation of motion for horizontal motion:
95m = v_x × t
Substituting the value of v_x from equation (1):
95m = (v × cos(θ)) × t
Now, we can substitute the value of t from equation (2):
95m = (v × cos(θ)) × √(2 × 100m / g)
Simplifying the equation:
95m = (v × cos(50.8°)) × √(200m / 9.8m/s^2)
95m = (v × cos(50.8°)) × √(20.41)
Now, we can solve this equation to find the value of v.