A cannon sends a projectile towards a target 1390m away. The initial velocity makes an angel of 39 degrees with the horizontal. The target is hit.
The acceleration of gravity is 9.8m/s^2. what is the magnitude of the initial velocity?
To find the magnitude of the initial velocity, we can apply the principles of projectile motion and use the given information.
Step 1: Identify the known values:
- Distance to the target (range): 1390 m
- Angle of the initial velocity with the horizontal: 39 degrees
- Acceleration due to gravity: 9.8 m/s^2
Step 2: Analyze the projectile motion:
In projectile motion, we can split the initial velocity into its horizontal and vertical components.
The horizontal component (Vx) remains constant throughout the motion since there is no force acting horizontally. The vertical component (Vy) changes due to the gravitational acceleration.
Step 3: Find the horizontal component of velocity:
Vx = V * cos(θ)
where V is the magnitude of the initial velocity and θ is the angle with the horizontal.
Vx = V * cos(39°)
Step 4: Find the time of flight:
The time taken for the projectile to reach the target can be found using the horizontal distance and the horizontal component of velocity.
Time of flight (T) = Distance / Horizontal velocity
T = 1390 m / V * cos(39°)
Step 5: Find the vertical component of velocity:
Taking the vertical direction as positive upwards, the initial vertical component of velocity (Vy) can be found using the equation:
Vy = V * sin(θ)
Step 6: Find the time to reach maximum height:
The projectile will reach its maximum height when Vy becomes zero. We can find the time taken to reach the maximum height (t) using the vertical component of velocity and the acceleration due to gravity.
0 = Vy - g * t
t = Vy / g
Step 7: Find the total time of flight:
The total time of flight can be calculated as twice the time to reach maximum height (t) since it takes the same amount of time to reach the maximum height and fall back down.
Total time of flight = 2 * t
Total time of flight = 2 * (Vy / g)
Step 8: Substitute the time of flight into the horizontal velocity equation:
1390 m / V * cos(39°) = 2 * (Vy / g)
Step 9: Substitute the vertical component of velocity using the equation:
1390 m / V * cos(39°) = 2 * (V * sin(39°) / g)
Step 10: Solve for V (magnitude of initial velocity):
V = (1390 m * g) / (2 * cos(39°) * sin(39°))
Calculating the above expression will give us the magnitude of the initial velocity. By substituting the given values for g, 1390 m, and the angle, you can find the final answer to the problem.