A projectile is fired from the top of a cliff at 14.2 m/s at an angle of 36.9 degrees below the horizontal, and strikes water at the same level as the base of the cliff 3.51 seconds later. (Assume g=10.0 m/s2.)
How high was the top of the cliff above the water?
all we care about is the downward velocity. Initially that is
14.2 cos -36.9° = -11.36 m/s
so, starting from height H, the height after t seconds is
-11.36t - 4.9 t^2
at t=3.51,
h = -100.24 m
So, the cliff is 100.24m high
To find the height of the top of the cliff above the water, we need to break down the projectile's motion into horizontal and vertical components. Let's start by analyzing the vertical motion.
We know that the projectile's initial vertical velocity is given by the equation:
Vy = V * sin(θ)
Where:
- Vy is the initial vertical velocity
- V is the initial velocity of the projectile (14.2 m/s)
- θ is the angle below the horizontal (36.9 degrees)
Given this information, we can calculate the initial vertical velocity:
Vy = 14.2 m/s * sin(36.9 degrees)
Vy ≈ 8.59 m/s (rounded to two decimal places)
Next, we can determine the time it takes for the projectile to reach its maximum height. The time t at which the projectile reaches its highest point can be calculated using the equation:
t = Vy / g
Where:
- g is the acceleration due to gravity (10.0 m/s^2)
Plugging in the values we know, we can calculate t:
t = 8.59 m/s / 10.0 m/s^2
t ≈ 0.859 seconds (rounded to three decimal places)
Since we know that the projectile hit the water 3.51 seconds after being fired, we can determine the total time of flight by doubling the time it took to reach the maximum height:
Total time of flight = 2 * t
Total time of flight = 2 * 0.859 seconds
Total time of flight ≈ 1.718 seconds (rounded to three decimal places)
Now that we have the total time of flight, we can find the vertical distance traveled by the projectile using the equation:
DeltaY = Vyt + (1/2) * (-g) * t^2
Where:
- DeltaY is the vertical distance traveled by the projectile
- Vy is the initial vertical velocity
- g is the acceleration due to gravity
- t is the total time of flight
Plugging in the values we know, we can calculate DeltaY:
DeltaY = 8.59 m/s * 1.718 seconds + (1/2) * (-10.0 m/s^2) * (1.718 seconds)^2
DeltaY ≈ 12.12 meters (rounded to two decimal places)
Finally, to find the height of the top of the cliff above the water, we subtract the initial height from the total vertical distance traveled by the projectile:
Height of the cliff = DeltaY - initial height
Height of the cliff = 12.12 meters - 0 meters (since the water is at the same level as the base of the cliff)
Height of the cliff ≈ 12.12 meters (rounded to two decimal places)
Therefore, the top of the cliff was approximately 12.12 meters above the water.