Fireworks are fired at an angle of 82degrees with the horizontal. They are expected to explode about 100 m in the air, 4.8 secs after they are fired. What is the initial velocity of a shell fired from ground level.
I think I have to find the vertical and horizontal components first but I don't know where to go after I do so. Thanks for any help.
Well, the formula to answer this is 100=-16(4.8)^2+VoSin(82°)4.8+0
You can find Vo based off of this but, it’s been 9 years so I don’t think you really need this anymore.
To find the initial velocity of the shell fired from ground level, you need to consider both the vertical and horizontal components of the projectile motion.
First, let's break down the initial velocity into its vertical and horizontal components. The initial velocity can be represented as:
v₀ = v₀x + v₀y
where v₀ is the initial velocity, v₀x represents the horizontal component of the velocity, and v₀y represents the vertical component of the velocity.
Since the firework is fired at an angle of 82 degrees with the horizontal, we can use trigonometric functions to find the vertical and horizontal components of the velocity.
The vertical component, v₀y, can be found using the formula:
v₀y = v₀ * sin(θ)
where θ is the angle of projection.
The horizontal component, v₀x, can be found using the formula:
v₀x = v₀ * cos(θ)
From the problem, we are given that the firework is expected to explode at a height of 100 m and the time taken to reach that height is 4.8 seconds.
Using the equation of motion for the vertical displacement:
Δy = v₀y * t + (1/2) * a * t²
Since we are considering vertical motion, the acceleration, a, can be taken as the acceleration due to gravity, which is approximately -9.8 m/s² (negative value as it acts in the opposite direction to the upward direction).
Substituting the given values, we have:
100 = v₀ * sin(82°) * 4.8 + (1/2) * (-9.8) * (4.8)²
Now, we can solve this equation for v₀.
Similarly, the horizontal displacement can be calculated using the formula:
Δx = v₀x * t
Since the horizontal displacement is not given, we can assume it to be zero (as fired from ground level).
Substituting the given values, we have:
0 = v₀ * cos(82°) * 4.8
Now, we can solve this equation for v₀.
By solving these two equations simultaneously, you can find the initial velocity, v₀, of the shell fired from ground level.