a pelican flying a horizontal path drops a fish from a height of 5.0m. the fish travels 8.1m horizontaly before it hits the water below. what was the pelicans initial speed?
To find the pelican's initial speed, we can use the principles of projectile motion. When the fish is dropped, it becomes a projectile moving horizontally under the influence of gravity.
We can assume that there is no air resistance, so the horizontal velocity of the fish remains constant throughout its motion. The vertical motion of the fish is solely due to the acceleration due to gravity.
Here's how we can calculate the pelican's initial speed:
1. Determine the vertical displacement of the fish:
The fish falls from a height of 5.0m, so its vertical displacement is 5.0m.
2. Use the formula for vertical displacement in projectile motion:
The formula is: Δy = v₀y * t + (1/2) * g * t²,
where Δy is the vertical displacement, v₀y is the initial vertical velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s²).
Since the initial vertical velocity of the fish is zero (dropped vertically), the equation becomes:
5.0m = 0 * t + (1/2) * (-9.8 m/s²) * t²,
which simplifies to:
5.0m = -4.9 m/s² * t².
3. Solve for the time (t):
Rearranging the equation for t² gives us:
t² = (5.0m) / (-4.9 m/s²).
Taking the square root of both sides, we find:
t ≈ √(5.0m / (-4.9 m/s²)).
The square root of a negative value doesn't have a real solution, so we can take the absolute value of the time:
t ≈ √(5.0m / 4.9 m/s²).
4. Calculate the horizontal velocity (v₀x):
Since the horizontal velocity is constant, we can find it using the formula:
v₀x = (horizontal displacement) / t.
In this case, the horizontal displacement is 8.1m, and we found the value of t in step 3.
Substituting the values, we have:
v₀x ≈ 8.1m / √(5.0m / 4.9 m/s²).
5. Calculate the initial speed (v₀):
The initial speed is the magnitude of the velocity vector, given by:
v₀ = √(v₀x² + v₀y²).
Since the fish is dropped vertically, its initial vertical velocity (v₀y) is zero.
The equation becomes:
v₀ = √(v₀x² + 0).
Substitute the value of v₀x found in step 4 and calculate the result.
This will give you the pelican's initial speed.