A dolphin leaps out of the water at an angle of 24.9 ° above the horizontal. The horizontal component of the dolphin's velocity is 5.05 m/s. Find the magnitude of the vertical component of the velocity.
To find the magnitude of the vertical component of the dolphin's velocity, we can use trigonometry.
First, let's label the horizontal component as the adjacent side (A) and the vertical component as the opposite side (O) in a right-angled triangle.
Given:
Angle (θ) = 24.9°
Horizontal component (A) = 5.05 m/s
We can use the following trigonometric identity:
sin(θ) = O / H
In this case, H represents the hypotenuse of the right-angled triangle, which represents the magnitude of the dolphin's velocity.
Since we only know the value of A (the adjacent side), we can use the following trigonometric identity:
cos(θ) = A / H
Rearranging the equation to solve for H, we get:
H = A / cos(θ)
Substituting the values into the equation:
H = 5.05 m/s / cos(24.9°)
Calculating the value of cos(24.9°) using a calculator, we find it to be approximately 0.9093.
H = 5.05 m/s / 0.9093
H ≈ 5.54 m/s
Therefore, the magnitude of the vertical component of the dolphin's velocity is approximately 5.54 m/s.
To find the magnitude of the vertical component of the velocity, we can use trigonometry. The vertical component of the velocity can be found by using the sine of the angle of 24.9°.
First, let's define the given variables:
θ = 24.9° (angle above the horizontal)
V_horizontal = 5.05 m/s (horizontal component of velocity)
To find the vertical component of the velocity (V_vertical), we can use the formula:
V_vertical = V_total * sin(θ)
where V_total is the total velocity of the dolphin and sin(θ) is the sine of the angle θ.
To calculate V_total, we can use the Pythagorean theorem, which states that the square of the hypotenuse (V_total) is equal to the sum of the squares of the two other sides (V_horizontal^2 + V_vertical^2). Rearranging the formula, we get:
V_total = sqrt(V_horizontal^2 + V_vertical^2)
Now we can substitute V_total into the equation for V_vertical:
V_vertical = sqrt(V_total^2 - V_horizontal^2) * sin(θ)
Plugging in the given values:
V_vertical = sqrt((5.05 m/s)^2 - (V_horizontal^2)) * sin(24.9°)
Calculating the magnitude of the vertical component of the velocity will give you the answer to the problem.