projectiles what is the formula?
Question A firework company is designing fireworks so that they can explode above a certain height. the fireworks are fired from the ground at an initial velocity of 160 ft per sec, what is the range of times for the firework to explode so that it is at least 350 feet in the air
To find the range of times for the firework to explode, we need to understand the projectile motion and use the relevant formula.
Projectile motion occurs when an object is projected into the air with an initial velocity and follows a curved trajectory under the influence of gravity.
In this case, the firework is fired vertically from the ground, so only the vertical component of the motion is of interest. The horizontal component will not affect the height reached by the firework.
The formula to determine the height of a projectile at a given time is:
h = h0 + v0y * t - 0.5 * g * t^2
Where:
- h is the height of the projectile above the ground
- h0 is the initial height of the projectile (0 in this case, as it is fired from the ground)
- v0y is the vertical component of the initial velocity (in this case, 160 ft/s)
- t is the time elapsed since the projectile was launched
- g is the acceleration due to gravity (approximately 32.2 ft/s^2)
To find the range of times for the firework to reach at least 350 feet, we need to solve the formula for t:
350 = 0 + 160 * t - 0.5 * 32.2 * t^2
This equation is a quadratic equation where we need to find the times (t) that satisfy it.
Now, we can use various methods to solve the quadratic equation. One common approach is to use the quadratic formula. The quadratic formula is:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation:
a = -0.5 * 32.2
b = 160
c = -350
Substituting these values into the formula, we can calculate the roots (values of t) that correspond to the height being at least 350 feet.
By plugging the values of a, b and c into the quadratic formula and solving for t, we can find the range of times for the firework to explode at least 350 feet in the air.