You are standing at the top of a cliff that has a stair step configuration. There is a vertical drop of 7 m at your feet, then a horizontal shelf of 8 m, then another drop of 3 m to the bottom of the canyon, which has a horizontal floor. You kick a rock, giving it an initial horizontal velocity that barely clears the 8 m shelf below.
What initial horizontal velocity v will be required to barely clear the edge of the shelf below you? The acceleration of gravity is 9.8 m/s2 . Consider air friction to be negligible.
Answer in units of m/s
I use u for horizontal and v for vertical
first do a 7 m vertical fall
v = g t
h = (1/2) v t^2
7 = 4.9 t^2
t = 1.2 sec
so it must go horizontal 8 m in 1.2 s
so
u = 8/1.2 = 6.7 m/s horizontal
Well, that sounds like quite a precarious situation! We need to find the initial horizontal velocity, v, required to barely clear the edge of the shelf below.
Let's break it down step by step. First, we have a vertical drop of 7 m, meaning the rock will fall for a certain amount of time before reaching the horizontal shelf.
Using the equation for the vertical motion of an object under constant acceleration, h = (1/2) * g * t^2, where h is the distance fallen (7 m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time of fall.
We rearrange the equation to solve for time: t = sqrt((2h) / g).
Plugging in the given values: t = sqrt((2 * 7) / 9.8) ≈ 1.50 s.
Now that we know the time it takes for the rock to fall, we can calculate the horizontal distance of the shelf it will cover. We use the equation d = v * t, where d is the horizontal distance (8 m) and t is the time of fall (1.50 s).
Rearranging the equation to solve for the initial horizontal velocity: v = d / t.
Plugging in the given values: v = 8 / 1.50 ≈ 5.33 m/s.
So, the initial horizontal velocity required to barely clear the edge of the shelf below is approximately 5.33 m/s.
But hey, rock on with your calculations!
To find the initial horizontal velocity required to barely clear the edge of the shelf below, we first need to find the time it takes for the rock to fall from the initial vertical drop of 7 m to the lower drop of 3 m.
1. Use the equation for the time of flight for an object thrown vertically upwards or downwards:
t = √(2h / g)
where t is the time of flight, h is the vertical distance, and g is the acceleration due to gravity.
2. For the first drop of 7 m, the time of flight is:
t1 = √(2 * 7 / 9.8) = √1.42857 ≈ 1.195 s
3. Now, we need to find the horizontal distance covered by the rock during this time, which is the horizontal distance of the shelf (8 m).
4. To determine the initial horizontal velocity, use the formula:
v = d / t
where v is the initial horizontal velocity, d is the horizontal distance, and t is the time of flight.
5. Substituting the values:
v = 8 m / 1.195 s = 6.690 m/s
Therefore, the initial horizontal velocity required to barely clear the edge of the shelf is approximately 6.690 m/s.
To determine the initial horizontal velocity required to barely clear the edge of the shelf below, we can use the principles of projectile motion.
First, let's analyze the vertical motion of the rock. From the information given, we have two vertical drops: one of 7 m and another of 3 m. The total vertical distance the rock needs to cover is 10 m (7 m + 3 m).
Using the kinematic equation for vertical motion:
Δy = v_iy * t + (1/2) * a * t^2
Where:
Δy = change in vertical distance (10 m)
v_iy = initial vertical velocity (which is 0, since the rock is only given an initial horizontal velocity)
a = acceleration due to gravity (-9.8 m/s^2)
t = time of flight
Since the rock falls freely under gravity, we can rearrange the equation to solve for the time of flight:
t = sqrt((2 * Δy) / a)
Plugging in the values:
t = sqrt((2 * 10 m) / (-9.8 m/s^2))
Calculating the time of flight, we find:
t ≈ 1.43 seconds
Now, let's analyze the horizontal motion of the rock. The horizontal distance the rock needs to cover before it drops is 8 m.
Using the equation:
Δx = v_ix * t
where Δx = horizontal distance (8 m), v_ix = initial horizontal velocity, and t = time of flight
Rearranging the equation to solve for the initial horizontal velocity:
v_ix = Δx / t
Plugging in the values:
v_ix = 8 m / 1.43 s
Calculating the initial horizontal velocity, we find:
v_ix ≈ 5.59 m/s
Therefore, the initial horizontal velocity required to barely clear the edge of the shelf below is approximately 5.59 m/s.