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ME = 13520 joules + 26950 joules = 40470 joules
Validation
Let's see if we can validate that result in some other way.
We know that the mechanical energy of the rocket at rest on the platform was equal to 1470 joules.
We can compute the work done in moving the rocket up by 260 meters by multiplying that distance by the upward force. Thus,
work = distance * thrust, or
work = 260 m * 150N = 39000 joules
Additional mechanical energy
This is the mechanical energy added to the rocket after it left the platform while the engine was burning.The total mechanical energy when the burn ends is the sum of that value and the mechanical energy that it had while at rest on the platform. Thus, the totalmechanical energy at the end of the burn is:
ME = 1470 joules + 39000 joules = 40470 joules
A good match
This value matches the value computed earlier on the basis of the height of the rocket above the surface of the earth and thevelocity of the rocket. Thus, the two approaches agree with one another up to this point.
State at the end of Leg B
Therefore, at the end of Leg B,
This is the part of the trip where the rocket coasts from its height at the end of Leg B to the apex of its trip. The continued upward motion is due solely to its kinetic energy at the endof Leg B.
From the end of Leg B when the rocket engine stops burning, until the rocket crashes on the surface of the earth, the only forces acting on the rocket will be the internal force of gravity.
Total mechanical energy is conserved
Since internal forces cannot change the mechanical energy possessed by an object, thetotal mechanical energy for the rocket must remain at 40470 joules for the remainder of the trip.
How long to reach the apex?
We can compute the time required for the rocket to reach the apex as
t = v/g = (52m/s)/(9.8m/s^2) = 5.31 seconds
How far will the rocket travel?
Knowing the time required to reach the apex, we can compute the distance to the apex (during this leg only) as
d = v0*t - 0.5*g*t^2, or
d = (52m/s) * (5.31s) - (0.5) * (9.8m/s^2)*(5.31s)^2, or
d = 138 meters
An additional 138 meters
In other words, the rocket travels an additional 138 meters straight up after the rocket-engine stops burning. This additional travel is due solely to thekinetic energy possessed by the rocket at the end of the burn.
The total height of the apex
Adding 138 more meters to the height at the end of the burn causes the height at the apex to be
height at apex = 275m + 138m = 413 meters
Mechanical energy equals potential energy alone
At that point, the total mechanical energy is equal to the gravitational potential energy because the rocket isn't moving and the kinetic energy has gone to zero.
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