The vertical and horizontal component of the initial velocity of a projectile are 36m/s and 64m/s. Find the initial velocity and angle inclination
To find the initial velocity and angle of inclination of a projectile, we can use the following trigonometric relationships:
The horizontal component of the initial velocity (Vx) can be found using the equation:
Vx = V * cos(theta)
where V is the magnitude of the initial velocity and theta is the angle of inclination.
Similarly, the vertical component of the initial velocity (Vy) can be found using the equation:
Vy = V * sin(theta)
Given that Vx = 36 m/s and Vy = 64 m/s, we can solve these equations simultaneously to find V and theta.
We can start by rearranging the Vy equation to solve for V:
V = Vy / sin(theta)
Substituting the given values:
V = 64 m/s / sin(theta)
Next, we can substitute this expression for V in the Vx equation:
36 m/s = (64 m/s / sin(theta)) * cos(theta)
Next, we can simplify this equation by multiplying both sides by sin(theta):
36 m/s * sin(theta) = 64 m/s * cos(theta)
Now, cancel out the m/s terms:
36 * sin(theta) = 64 * cos(theta)
Next, rearrange the equation to separate the terms involving theta on one side:
36 * sin(theta) - 64 * cos(theta) = 0
Now, we can use trigonometric identities to simplify this equation. Specifically, we can use the identity sin(theta - phi) = sin(theta) * cos(phi) - cos(theta) * sin(phi).
In this case, we can rewrite the equation as:
36 * sin(theta) - 64 * cos(theta) = 0
36 * sin(theta) - 64 * sin(theta - 90 degrees) = 0
Simplifying further:
36 * sin(theta) - 64 * sin(theta) * cos(90 degrees) - 64 * cos(theta) * sin(90 degrees) = 0
36 * sin(theta) - 64 * sin(theta) * 0 - 64 * cos(theta) * 1 = 0
36 * sin(theta) - 64 * cos(theta) = 0
Rearranging this equation:
36 * sin(theta) = 64 * cos(theta)
Dividing both sides by cos(theta):
(36 * sin(theta)) / cos(theta) = 64
Using the tangent identity tan(theta) = sin(theta) / cos(theta), we can rewrite this equation as:
36 * tan(theta) = 64
Now, we can solve for theta:
tan(theta) = 64 / 36
theta = arctan(64 / 36)
Using a calculator, we find:
theta ≈ 59.04 degrees
Now that we have the angle of inclination, we can substitute it into the earlier equation to solve for V:
V = Vy / sin(theta)
V = 64 m/s / sin(59.04 degrees)
Using a calculator, we find:
V ≈ 72.08 m/s
Therefore, the initial velocity of the projectile is approximately 72.08 m/s and the angle of inclination is approximately 59.04 degrees.
To find the initial velocity and angle of inclination of a projectile with given vertical and horizontal components of initial velocity, we can use basic trigonometry.
Let's assume the vertical component of the initial velocity is Vx and the horizontal component is Vy.
Given:
Vertical component (Vx) = 36 m/s
Horizontal component (Vy) = 64 m/s
To find the initial velocity (V) and angle of inclination (θ), we can use the following formulas:
1. V = √(Vx^2 + Vy^2)
2. θ = atan(Vy / Vx)
Step 1: Calculate the initial velocity (V):
V = √(36^2 + 64^2)
V = √(1296 + 4096)
V = √5392
V ≈ 73.49 m/s
Step 2: Calculate the angle of inclination (θ):
θ = atan(64 / 36)
θ = atan(1.78)
θ ≈ 62.99°
Therefore, the initial velocity of the projectile is approximately 73.49 m/s, and the angle of inclination is approximately 62.99°.
Vi = 36
u = 64
tan A = 36/64
|v| = sqrt (36^2 + 64^2)
= 4 sqrt(337)