Write an equation that represents the path of a firework that is launched from the ground with the initial velocity that you chose.
(Choose whatever initial velocity and consider where that fits into the equation… this time you are launching the firework from the ground so consider how that changes the equation)
To represent the path of a firework launched from the ground with an initial velocity, we can use the equation for vertical motion.
Let's assume the firework is launched upward with an initial velocity of v₀ m/s:
The equation for vertical motion is:
h(t) = v₀t - (1/2)gt²
Where:
- h(t) represents the height of the firework at time t,
- v₀t represents the initial upward velocity multiplied by time,
- g represents the acceleration due to gravity (approximately 9.8 m/s²),
- (1/2)gt² represents the effect of gravity pulling the firework downward over time.
Since we launched the firework from the ground, its initial height is 0. Therefore, we can simplify the equation as:
h(t) = v₀t - (1/2)gt²
This equation represents the path of a firework launched from the ground with an initial velocity of v₀.
To represent the path of a firework launched from the ground, we can use the equations of motion for projectile motion.
The equation for the vertical displacement of the firework, assuming no air resistance, is given by:
y = y₀ + (v₀y * t) - (1/2 * g * t²)
where:
y is the vertical displacement of the firework,
y₀ is the initial vertical position (which is 0 since the firework is launched from the ground),
v₀y is the initial vertical velocity (which we need to choose),
t is the time elapsed, and
g is the acceleration due to gravity (approximately 9.8 m/s²).
Since the firework is launched from the ground, the initial vertical position (y₀) is 0, and the equation simplifies to:
y = v₀y * t - (1/2 * g * t²)
This equation gives us the vertical position of the firework (y) at any given time (t) when launched from the ground with an initial vertical velocity (v₀y) that we choose.
To represent the path of a firework launched from the ground with an initial velocity, we can use the equation of motion for vertical projectile motion:
𝑦(𝑡) = 𝑦₀ + 𝑣₀𝑡 - 1/2𝑔𝑡²
In this equation, 𝑦(𝑡) represents the vertical position (height) of the firework at time 𝑡, 𝑦₀ represents the initial vertical position (which is zero in this case since we're launching it from the ground), 𝑣₀ represents the initial vertical velocity, 𝑔 represents the acceleration due to gravity (which is approximately 9.8 m/s²), and 𝑡 represents time.
So, the equation representing the path of the firework would be:
𝑦(𝑡) = 𝑣₀𝑡 - 1/2𝑔𝑡²