find a projectile fired with an initial velocity of 128 feet per second at an angle of 26 degrees
I assume I will find it where it landed. That would be where the height is zero.
y = tanθ x - g/(2(vcosθ)^2) x^2
In this case, that's
y = 0.4877x - 0.0012088x^2
y=0 at x=403.5
So, I'd start looking at 403.5 feet from where it was shot.
Did you have something else in mind when you said to find it?
To find the projectile's time of flight, maximum height, horizontal range, and impact velocity, we can analyze the motion of the projectile along the x-axis and y-axis separately.
1. Analysis along the x-axis:
The initial velocity in the x-direction (Vx) can be found using the formula:
Vx = V * cos(θ)
where V is the initial velocity of 128 feet per second, and θ is the angle of 26 degrees.
Plugging in the values:
Vx = 128 * cos(26°)
Vx ≈ 115.54 feet per second
2. Analysis along the y-axis:
The initial velocity in the y-direction (Vy) can be found using the formula:
Vy = V * sin(θ)
where V is the initial velocity of 128 feet per second, and θ is the angle of 26 degrees.
Plugging in the values:
Vy = 128 * sin(26°)
Vy ≈ 55.62 feet per second
3. Time of flight:
The time of flight (T) can be determined using the formula:
T = (2 * Vy) / g
where g is the acceleration due to gravity (32.2 feet per second squared).
Plugging in the values:
T = (2 * 55.62) / 32.2
T ≈ 3.46 seconds
4. Maximum height:
The maximum height (H) can be found using the formula:
H = (Vy^2) / (2 * g)
Plugging in the values:
H = (55.62^2) / (2 * 32.2)
H ≈ 47.74 feet
5. Horizontal range:
The horizontal range (R) can be calculated using the formula:
R = Vx * T
Plugging in the values:
R = 115.54 * 3.46
R ≈ 399.85 feet
6. Impact velocity:
Since there is no air resistance, the impact velocity along the y-axis (Vy') is equal to the initial velocity along the y-axis (-Vy):
Vy' = -Vy ≈ -55.62 feet per second
The impact velocity along the x-axis (Vx') remains the same as the initial velocity along the x-axis (Vx):
Vx' = Vx ≈ 115.54 feet per second
Therefore, the projectile fired with an initial velocity of 128 feet per second at an angle of 26 degrees will have a time of flight of approximately 3.46 seconds, reach a maximum height of around 47.74 feet, and travel a horizontal range of approximately 399.85 feet. Upon impact, the velocity would be approximately -55.62 feet per second along the y-axis and 115.54 feet per second along the x-axis.