At the Earth's surface, a projectile is launched straight up at a speed of 8.4 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.

Set the initial kinetic energy equal to the potential energy change at the highest elevation, and solve for that elevation.

(1/2)mV^2 = GMm/R - GMm/(R + h)

Solve for the height h. m cancels out
M is the mass of the earth, G is the universal gravity constant and R is the earth's radius.

Well, it looks like someone's trying to shoot for the stars! Or at least, the atmosphere. Anyway, if you want to figure out how high this projectile will rise, we can use some handy-dandy physics.

First things first, let's convert that speed into meters per second because the metric system is the real MVP here. So, 8.4 km/s is equivalent to 8400 m/s. Impressive velocity, I must say!

Now, let's consider the motion of this projectile. Since it's going straight up, we only need to worry about its vertical velocity component. Gravity will work its magic and eventually bring our little buddy to a halt before it starts falling back down. At that point, its final velocity will be zero.

Now, there's a neat little formula called the kinematic equation that can help us find the height it will reach. It goes like this:

v² = u² - 2as,

where:
v is the final velocity (which is 0 m/s because it stops at the top),
u is the initial velocity (8400 m/s),
a is the acceleration (which is -9.8 m/s², since gravity is pulling it down), and
s is the displacement (the height we're trying to find).

Plugging in the values, we get:

0² = (8400)² - 2(-9.8)s,

which simplifies to:

0 = 70560000 - 19.6s.

Now, we can solve for s:

19.6s = 70560000,
s = 70560000 / 19.6.

Calculating that, we find that the projectile will rise approximately 3,600,000 meters (or 3,600 kilometers) high!

That's quite the climb! Just remember, once you reach the top, don't forget to take some cool selfies with the stars before gravity brings you back to Earth. Have a blast reaching for the sky!

To determine the height the projectile will reach, you can use the principles of projectile motion. The key equation to use in this case is the formula for the maximum height reached by an object in free fall.

The formula for calculating the maximum height reached by a projectile is given by:

H = (v^2)/(2g)

Where:
H = maximum height reached
v = initial velocity of the projectile
g = acceleration due to gravity (approximately 9.8 m/s^2)

First, we need to convert the initial velocity from km/s to m/s:

Initial velocity = 8.4 km/s = 8.4 * 1000 m/s = 8400 m/s

Now, we can plug the values into the formula:

H = (8400^2) / (2 * 9.8)
H = 7,056,000 / 19.6
H = 360,000 m

Therefore, the projectile will rise to a height of 360,000 meters (or 360 kilometers).

To determine the height reached by the projectile, we can use the principles of kinematics and the equations of motion. The projectile is launched straight up, which means it will decelerate due to gravity until it eventually reaches its maximum height before falling back down.

We need to find the maximum height achieved by the projectile, so we'll use the equation for vertical displacement:

Δy = V₀² / (2g)

Where:
Δy = Maximum height (vertical displacement)
V₀ = Initial velocity
g = Acceleration due to gravity (approximately 9.8 m/s²)

First, let's convert the initial velocity from km/s to m/s:
V₀ = 8.4 km/s * 1000 m/km = 8400 m/s

Now we can substitute the values into the equation:
Δy = (8400 m/s)² / (2 * 9.8 m/s²)

Calculating this expression, we find:
Δy = 71,285,714.29 m

So, the projectile will rise to a height of approximately 71,285,714.29 meters above the Earth's surface.