A Chinook salmon can jump out of water with a speed of 6.60 m/s . How far horizontally d can a Chinook salmon travel through the air if it leaves the water with an initial angle of θ=27.0° with respect to the horizontal? (Let the horizontal direction the fish travels be in the +x direction, and let the upward vertical direction be the +y direction. Neglect any effects due to air resistance.)
To determine the horizontal distance a Chinook salmon can travel through the air, we can use the principles of projectile motion.
Given:
Initial speed (v0) = 6.60 m/s
Launch angle (θ) = 27.0°
Using the horizontal and vertical components of velocity, we can calculate the horizontal distance (d) traveled. The horizontal component (v0x) and the vertical component (v0y) are related to the initial speed and launch angle as follows:
v0x = v0 * cos(θ)
v0y = v0 * sin(θ)
Substituting the given values into these equations:
v0x = 6.60 m/s * cos(27.0°)
v0y = 6.60 m/s * sin(27.0°)
v0x ≈ 5.9992 m/s
v0y ≈ 3.0156 m/s
Since we are neglecting air resistance, the time of flight is determined solely by the vertical motion. The formula for time of flight (t) is obtained from the equation:
v0y = gt
where g is the acceleration due to gravity (approximately 9.8 m/s²). Rearranging the equation, we can solve for t:
t = v0y / g
Substituting the known values:
t ≈ 3.0156 m/s / 9.8 m/s²
t ≈ 0.3079 s
Finally, we can find the horizontal distance (d) traveled by the Chinook salmon using the formula:
d = v0x * t
Substituting the known values:
d ≈ 5.9992 m/s * 0.3079 s
d ≈ 1.8437 m
Therefore, a Chinook salmon can travel approximately 1.8437 meters horizontally through the air.
To find the horizontal distance traveled by the Chinook salmon, we need to break down the initial velocity (6.60 m/s) into its horizontal (x) and vertical (y) components.
Given:
Initial velocity magnitude, V = 6.60 m/s
Launch angle with respect to horizontal, θ = 27.0°
Horizontal component, Vx = V * cos(θ)
Vertical component, Vy = V * sin(θ)
Let's calculate these components:
Vx = 6.60 m/s * cos(27.0°)
Vx = 6.60 m/s * 0.891
Vx = 5.87 m/s
Vy = 6.60 m/s * sin(27.0°)
Vy = 6.60 m/s * 0.454
Vy = 2.99 m/s
Now, we can use the horizontal component (Vx) to calculate the horizontal distance traveled (d).
The time taken for the salmon to reach the maximum height (when Vy = 0) is given by:
t = Vy / g
where g is the acceleration due to gravity (9.8 m/s^2).
t = 2.99 m/s / 9.8 m/s^2
t ≈ 0.305 s
The horizontal distance traveled can be found using the formula:
d = Vx * t
d = 5.87 m/s * 0.305 s
d ≈ 1.79 meters
Therefore, a Chinook salmon can travel approximately 1.79 meters horizontally through the air.