What is the missile's speed at the peak of its trajectory?

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In summary, the trajectory of a missile reaches a maximum altitude of 1200 km. Using the equation K+U = K_o+U_o and taking into account the mass of Earth, universal gravity, and the distance between the object and the center of Earth, the speed of the missile at the peak of its trajectory is approximately 24m/s. However, the calculation should also consider the distance to the center of Earth, which is 1200 * 1000m + Earth's radius.
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tnutty
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Homework Statement



A missile's trajectory takes it to a maximum altitude of 1200 km.
If its launch speed is 6.2 km/s , how fast is it moving at the peak of its trajectory?

Homework Equations



K+U = K_o+U_o

U(r) = -GMm/r

The Attempt at a Solution



Alright some people have been telling me its zero. Intuitively that's what I thought, but
I guess its wrong. Nevertheless, here is my attempt :

K+U = K_o +U_o;

=

1/2mV^2 - GM_eM/r_d = 1/2mV_o^2 - GM_em / r_e

and solving for
V^2 = V_o^2 - 2GM_e/r_e + GM_e/r_d

where,
V_o = 6.2km -> 6.2*1000m
G = universal graity = 6.67*10^-11
M_e = mass of Earth = 5.98*10^24
r_e = radius of Earth = 6.37 * 10^6;
r_d = distance between the object and the center of earth
= 1200km -> 1200*1000m

and my answer is terms of meters
~2400m

so dividing by 1000

and

v_p ~ 24m/s
 
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  • #2


tnutty said:
Alright some people have been telling me its zero. Intuitively that's what I thought, but
I guess its wrong.
Why would you think that? (It may be moving horizontally at the peak.)

Nevertheless, here is my attempt :

K+U = K_o +U_o;

=

1/2mV^2 - GM_eM/r_d = 1/2mV_o^2 - GM_em / r_e
Looks good.

and solving for
V^2 = V_o^2 - 2GM_e/r_e + GM_e/r_d
You left off a factor of 2 in that last term.

where,
V_o = 6.2km -> 6.2*1000m
G = universal graity = 6.67*10^-11
M_e = mass of Earth = 5.98*10^24
r_e = radius of Earth = 6.37 * 10^6;
r_d = distance between the object and the center of earth
= 1200km -> 1200*1000m
1200km is just the altitude, not the distance to the center of the earth.
 
  • #3


so is this correct?

Although my formula missed a factor of 2, I still did the calculation with the 2 factored in.
Can you check my calculations?
 
  • #4


Did you correct your value of r_d?
 
  • #5


no, I am not sure why I should. The units won't match.

I converted everything into meter and the final answer into km
 
  • #6


The issue is not units, but that you are using r_d = 1200 km. That's the altitude; you need the distance to the center of the earth.
 
  • #7


so is the distance 1200km + Earth's radius?
 
  • #8


how do I get the distance to the center of the Earth of the object?
 
  • #9


tnutty said:
so is the distance 1200km + Earth's radius?
Yes, that's r_d.
tnutty said:
how do I get the distance to the center of the Earth of the object?
See above.
 
  • #10


So the distance is 1200 * 1000m + Earth's radius
because
1) It they are in both meters.

2) Because the projectory is launched from the Earth's surface, or top of the earth?
 
  • #11


Good
 
  • #12


Cool, thanks!
 

1. What is a missile's trajectory problem?

The missile's trajectory problem is a mathematical problem that involves calculating the flight path of a missile from its launch point to its intended target. It takes into account variables such as the missile's initial velocity, acceleration, and the effects of air resistance to determine the path it will take.

2. Why is the missile's trajectory problem important?

The missile's trajectory problem is important because it allows scientists and engineers to accurately predict the path of a missile and ensure it reaches its intended target. It is also crucial for developing and improving defense systems against missiles.

3. What factors affect a missile's trajectory?

Several factors can affect a missile's trajectory, including its initial velocity, the force of gravity, air resistance, and the rotation of the Earth. External factors such as wind and temperature can also play a role in altering the missile's path.

4. How is the missile's trajectory calculated?

The missile's trajectory is calculated using mathematical equations such as projectile motion equations and differential equations. These equations take into account the initial conditions of the missile, such as its velocity and angle of launch, as well as the forces acting upon it during its flight.

5. What are the challenges in predicting a missile's trajectory?

The main challenges in predicting a missile's trajectory include accounting for external factors such as wind and air resistance, as well as accurately measuring the missile's initial conditions. The complexity of the calculations involved also presents a challenge, requiring advanced mathematical and computational skills.

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