A projectile was fired out of a lead ejector at an angle of 45 degrees up from horizontal. The projectile velocity was 1000.00 meters per second. How high did it go? and how far did the projectile go?
initial speed up Vi = 1000 sin 45 = 707 m/s
v = Vi - 9.81 t
when v = 0, we go no higher
t = 707/9.81 = 72 seconds upward
h = Vi t - (1/2) 9.81 t^2
h = 707 (72) - 4.9 * 5184
= 50904-25427
25,000 m
u = horizontal speed = 1000cos 45 = 707
u t = 707 * 2*72 = 102,000 m
To determine how high the projectile went and how far it traveled, we can use the equations of projectile motion. First, let's break down the initial velocity into its horizontal and vertical components.
1. Horizontal velocity (Vx):
Since the projectile was fired at a 45-degree angle, the horizontal component of the velocity can be found using the formula Vx = V * cos(θ), where V is the initial velocity and θ is the angle.
Vx = 1000.00 m/s * cos(45°)
Vx = 1000.00 m/s * 0.7071
Vx ≈ 707.1 m/s
2. Vertical velocity (Vy):
The vertical component of the velocity can be found using the formula Vy = V * sin(θ).
Vy = 1000.00 m/s * sin(45°)
Vy = 1000.00 m/s * 0.7071
Vy ≈ 707.1 m/s
Now, let's find the time it takes for the projectile to reach its highest point (the time of flight).
3. Time of flight (t):
The time taken to reach the maximum height is equal to the time taken to return to the same vertical position, as the motion is symmetrical. At the apex, Vy = 0.
Using the equation Vy = Vy0 + a * t, where a is the acceleration due to gravity (-9.8 m/s²) and Vy0 is the initial vertical velocity:
0 = 707.1 m/s + (-9.8 m/s²) * t
Solving for t:
t = -707.1 m/s / -9.8 m/s²
t ≈ 72.096 seconds
Now, let's calculate the maximum height reached by the projectile.
4. Maximum height (h):
The vertical distance traveled can be calculated using the equation h = Vy0 * t + 0.5 * a * t².
h = 707.1 m/s * 72.096 s + 0.5 * (-9.8 m/s²) * (72.096 s)²
h ≈ 256109.2 meters
So, the projectile reached a height of approximately 256109.2 meters.
To find the horizontal distance traveled by the projectile, we can use the horizontal velocity and the time of flight.
5. Horizontal distance (d):
The horizontal distance traveled is given by d = Vx * t.
d = 707.1 m/s * 72.096 s
d ≈ 50979.1 meters
Therefore, the projectile traveled a horizontal distance of approximately 50979.1 meters.