the average speed of a cannonball shot out of a standard cannon is 112m/s. if the cannon is titled so the cannonball launches at a 40 degree angle, how far will the cannonball travel?
To determine the distance traveled by the cannonball, we can analyze the horizontal and vertical components separately.
The initial velocity in the horizontal direction (Vx) remains constant throughout the motion as there is no force acting in this direction. Therefore, Vx = initial velocity * cos(angle).
Given that the initial velocity is 112 m/s and the angle is 40 degrees, we have:
Vx = 112 * cos(40) ≈ 86 m/s.
The time of flight (t) can be calculated from the vertical component of the motion. The vertical velocity (Vy) changes due to the force of gravity, but it reaches zero at the highest point of the trajectory. Thus, we can determine the time taken to reach the maximum height and then multiply it by 2 to account for the complete round trip of the ball.
The initial vertical velocity (Vy₀) is given by Vy₀ = initial velocity * sin(angle).
Given that the angle is 40 degrees, we have:
Vy₀ = 112 * sin(40) ≈ 71.7 m/s.
The time taken to reach the maximum height can be found using the formula:
Vy = Vy₀ - g*t,
where g is the acceleration due to gravity (approximately 9.8 m/s²).
At the highest point, Vy equals zero, so 0 = Vy₀ - g*t.
Therefore, t = Vy₀ / g.
Using the calculated vertical initial velocity of 71.7 m/s and the acceleration due to gravity of 9.8 m/s², we find:
t = 71.7 / 9.8 ≈ 7.3 s.
Since the time of flight is the same for the upward and downward paths, the total flight time is given by 2t = 2 * 7.3 ≈ 14.6 s.
Finally, to determine the horizontal distance traveled (d), we can multiply the horizontal component of velocity (Vx) by the total flight time (2t):
d = Vx * 2t = 86 * 14.6 ≈ 1255.6 m.
Therefore, the cannonball will travel approximately 1255.6 meters.
To find the horizontal distance traveled by the cannonball, we can break down the initial velocity of the cannonball into its horizontal and vertical components.
The horizontal component remains constant throughout the motion, while the vertical component is affected by gravity.
The horizontal component of the velocity (Vx) can be found using the formula:
Vx = V * cos(θ)
where V is the initial velocity of the cannonball and θ is the launch angle.
Substituting the given values, we get:
Vx = 112 m/s * cos(40°)
Vx = 112 m/s * 0.766
Vx ≈ 85.792 m/s
Now, we can find the total time of flight (t) using the vertical component of the velocity.
The vertical component of the velocity (Vy) can be found using the formula:
Vy = V * sin(θ)
Substituting the given values, we get:
Vy = 112 m/s * sin(40°)
Vy = 112 m/s * 0.642
Vy ≈ 71.904 m/s
To find the total time of flight (t), we can use the equation:
t = (2 * Vy) / g
where g is the acceleration due to gravity, approximately 9.8 m/s².
Substituting the values, we get:
t = (2 * 71.904 m/s) / 9.8 m/s²
t ≈ 14.624 seconds
Now, we can find the horizontal distance traveled (d) using the formula:
d = Vx * t
Substituting the values, we get:
d = 85.792 m/s * 14.624 seconds
d ≈ 1254.38 meters
Therefore, the cannonball will travel approximately 1254.38 meters.
To find the distance traveled by the cannonball, we can use the projectile motion equations. Let's break it down step by step:
Step 1: Resolve the initial velocity into its vertical and horizontal components.
The initial velocity of the cannonball can be resolved into two components: one vertical, Vy and the other horizontal, Vx.
Vx represents the horizontal component of the velocity, and Vy represents the vertical component.
Given:
Initial velocity (V0) = 112 m/s
Launch angle (θ) = 40 degrees
Using trigonometry, we can calculate the vertical and horizontal components of the initial velocity:
Vx = V0 * cos(θ)
Vy = V0 * sin(θ)
Substituting the values:
Vx = 112 m/s * cos(40 degrees)
Vy = 112 m/s * sin(40 degrees)
Step 2: Calculate the time of flight.
The time of flight (T) is the total time the cannonball remains in the air. This can be calculated using the vertical component of initial velocity.
T = 2 * Vy / g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the value of Vy and g:
T = 2 * (112 m/s * sin(40 degrees)) / 9.8 m/s^2
Step 3: Find the horizontal distance traveled.
The horizontal distance covered by the cannonball can be calculated using the horizontal component of initial velocity and the time of flight.
Distance (d) = Vx * T
Substituting the values:
Distance (d) = (112 m/s * cos(40 degrees)) * (2 * (112 m/s * sin(40 degrees)) / 9.8 m/s^2)
Now, you can simplify the equation and calculate the distance traveled by the cannonball.