A newly established colony on the Moon launches a capsule vertically with an initial speed of 1.453 km/s. Ignoring the rotation of the Moon, what is the maximum height reached by the capsule? km
The answer key is 1040km. I have tried v^2/2g (1.4532km/s)^2/(2*.0162km/s^2) and get the answer as 65.1km but that is not the correct answer. Can somebody shows me how to get to the answer key?
g = 1.62 m/s^2 = 0.00162 not 0.0162 km/s^2
but I suspect you have other unit errors
1.453 km/s * 3600 = 5231 km/hr which is awful fast
I get 651 km using your data with the g repaired
our answer isn't even close to the answer key. Have other idea?
have you had calculus?
the launch speed and low moon gravity means that the capsule goes beyond the region where g is relatively constant
the force is related to the height by the universal gravitational equation ... f = G M m / (r + h)^2
force * height = energy ... ∫ de = ∫ f dh = 1/2 m v ^2
sorry, but my calculus is pretty rusty
I think Scott is aiming you in the right direction
F = G M m/r^2
potential energy = integral from infinity to r of F dr
take it at 0 at infinity so it gets negative as you move in
F dr = G M m int dr/r^2 = -G M m /r
so as you move from Ro the surface to R, the increase is
G M m (1/Ro - 1/R)
that = 1/2 m v^2
note that G M/Ro^2 = g
hey ... this works
the gravitational potential energy of a object at a distance r from a large body is ... G M m / r
the difference in the P.E. on the surface and at altitude is the K.E. at launch
(G M m / r) - [G M m / (r + h)] = 1/2 m v^2
solve for h ... might be easier to plug in numerical values at the beginning
To find the maximum height reached by the capsule, we can use the principles of projectile motion.
First, let's break down the initial velocity into its vertical component. Since the capsule is launched vertically, the initial vertical velocity is equal to the total initial velocity. In this case, the vertical velocity would be 1.453 km/s.
Next, we can use the equation of motion for vertical motion:
y = (V₀y * t) - (0.5 * g * t²),
where:
- y is the height or displacement,
- V₀y is the initial vertical velocity,
- g is the acceleration due to gravity, and
- t is the time.
To find the maximum height, we need to find the time it takes for the capsule to reach its highest point. At this point, the vertical velocity will be zero.
Using the equation for vertical velocity, we have:
Vfy = V₀y - (g * t),
where Vfy is the final vertical velocity (which is zero at the maximum height).
Rearranging and solving for t:
t = V₀y / g,
t = 1.453 km/s / 1.62 km/s²,
t ≈ 0.897 seconds.
Now we can substitute this value of t into the equation for height:
y = (1.453 km/s * 0.897 s) - (0.5 * 1.62 km/s² * (0.897 s)²),
Simplifying this equation will give us the maximum height:
y ≈ 1.305 km - 0.648 km,
y ≈ 0.657 km.
It seems that there was a calculation error in your initial attempt. The correct value for the maximum height reached by the capsule should be approximately 0.657 km, not 65.1 km. Therefore, the correct answer in the answer key (1040 km) seems to be incorrect based on the given information. It's possible that there might be additional factors or values that need to be considered to arrive at the answer key value.