An object is projected upward from the surface of the Earth with an initial speed of 4.6 km/s. Find the maximum height it reaches.

To find the maximum height reached by an object projected upward from the surface of the Earth, we can use the equations of motion and take into account the acceleration due to gravity.

The initial speed of the object is given as 4.6 km/s. We need to convert this into m/s since the standard unit of acceleration is meters per second squared (m/s^2). To convert km/s to m/s, we multiply by 1000 (since there are 1000 meters in a kilometer). So, the initial speed is 4600 m/s.

Since the object is projected upward, its initial velocity is positive, but its acceleration due to gravity is negative since it is acting in the opposite direction. The acceleration due to gravity on Earth is approximately -9.8 m/s^2.

To find the maximum height, we need to determine how long it takes for the object to reach its maximum height. We can use the equation of motion: vf = vi + at, where vf is the final velocity, vi is the initial velocity, a is acceleration, and t is time.

At the maximum height, the final velocity becomes zero because the object momentarily stops before falling back down. So, we have 0 = 4600 m/s + (-9.8 m/s^2)t. Solving for t, we get t = 4600 m/s / 9.8 m/s^2 ≈ 469.39 s.

Now that we know how long it takes for the object to reach its maximum height, we can find the displacement using the equation: d = vit + (1/2)at^2, where d is displacement.

At maximum height, the displacement is equal to the maximum height reached, so d = h. Plugging in the values, we have h = 4600 m/s × 469.39 s + (1/2)(-9.8 m/s^2)(469.39 s)^2.

Simplifying the equation, we get h ≈ 1,069,888 meters or 1,070 km.

Therefore, the maximum height reached by the object is approximately 1,070 kilometers.

To find the maximum height reached by the object, we can use the equations of motion.

The equation that relates the initial velocity (u), final velocity (v), acceleration (a), and displacement (s) is:

v^2 = u^2 + 2as

Since the object is projected upward from the surface of the Earth, the final velocity (v) at the maximum height will be 0 m/s. The initial velocity (u) is 4.6 km/s, which needs to be converted to m/s:

u = 4.6 km/s = 4.6 * 10^3 m/s

Plugging these values into the equation, we have:

0 = (4.6 * 10^3)^2 + 2 * a * s

Simplifying the equation:

0 = 21160 * 10^6 + 2 * a * s

Since the object is moving against gravity, the acceleration (a) is -9.8 m/s^2 (negative sign represents the opposite direction to the displacement). Now, we can rearrange the equation to solve for the displacement (s):

2 * a * s = -21160 * 10^6

s = (-21160 * 10^6) / (2 * -9.8)

Dividing and solving:

s ≈ 1.083 * 10^6 m

Therefore, the maximum height the object reaches is approximately 1.083 * 10^6 meters.