A stone is thrown at an angle of 30 degrees above the horizontal from the top of a cliff with an initial speed of 12 m/s. If it hits the ground 4.6 seconds later, what is the height of the cliff?
59.8 metres
h=hi=Vi*t-4.9t^2 where vi=12sin30, hf=0
hi=4.9t^2-6t and t=4.6 solve for hi
To find the height of the cliff, we can use the equations of motion:
1. The horizontal motion equation:
x = v * t * cos(theta)
2. The vertical motion equation:
y = v * t * sin(theta) - (1/2) * g * t^2
Where:
- x is the horizontal distance traveled by the stone.
- y is the vertical distance from the top of the cliff.
- v is the initial speed of the stone.
- t is the time of flight.
- theta is the angle of projection.
- g is the acceleration due to gravity.
Given:
- Initial speed (v) = 12 m/s
- Angle of projection (theta) = 30 degrees
- Time of flight (t) = 4.6 seconds
- Acceleration due to gravity (g) = 9.8 m/s^2
Let's first find the horizontal distance traveled by the stone (x):
x = v * t * cos(theta)
x = 12 * 4.6 * cos(30)
x = 12 * 4.6 * sqrt(3) / 2
x = 26.467 meters (approx.)
Now, let's find the vertical distance from the top of the cliff (y):
y = v * t * sin(theta) - (1/2) * g * t^2
y = 12 * 4.6 * sin(30) - (1/2) * 9.8 * (4.6)^2
y = 12 * 4.6 * 0.5 - 4.9 * 21.16
y = 27.66 - 103.684
y = -76.024 meters (approx.)
Since the stone falls to the ground (y = -76.024), the height of the cliff is the absolute value of the vertical distance:
Height of the cliff = |y|
Height of the cliff = |-76.024|
Height of the cliff = 76.024 meters (approx.)
Therefore, the height of the cliff is approximately 76.024 meters.
To find the height of the cliff, we can use the equations of motion for projectile motion.
First, let's break down the initial velocity of the stone into its horizontal and vertical components. The horizontal component, Vx, can be found using the equation:
Vx = initial speed * cos(angle)
Vx = 12 m/s * cos(30 degrees)
= 12 m/s * √3/2
= 6√3 m/s
The vertical component, Vy, can be found using the equation:
Vy = initial speed * sin(angle)
Vy = 12 m/s * sin(30 degrees)
= 12 m/s * 1/2
= 6 m/s
Now, let's analyze the vertical motion of the stone. We can use the following equation to find the height (h) of the cliff:
h = Vy * t + 0.5 * g * t^2
where:
- h is the height of the cliff
- Vy is the vertical component of the initial velocity (6 m/s)
- t is the time taken for the stone to hit the ground (4.6 seconds)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values:
h = 6 m/s * 4.6 s + 0.5 * 9.8 m/s^2 * (4.6 s)^2
= 27.6 m + 0.5 * 9.8 m/s^2 * 21.16 s^2
= 27.6 m + 102.96 m
= 130.56 m
Therefore, the height of the cliff is approximately 130.56 meters.