To determine the speed of the rocket upon impact on the ground, we need to consider the motion of the rocket during the burn phase and the unpowered flight phase separately.
During the burn phase, the rocket experiences constant upward acceleration. We can use the kinematic equation:
v = u + at
Given that the rocket rises to 73 m, we know that the final displacement (y) is 73 m. At the instant of engine burnout, the rocket's velocity (v) is unknown, and the initial velocity (u) is also unknown. However, we know that the acceleration (a) is constant. Thus, we need to determine the value of acceleration.
To find the acceleration, we can use another kinematic equation:
v^2 = u^2 + 2as
At the end of the burn phase, the velocity becomes 0 m/s (since the engine burnout). The initial velocity (u) is also unknown. The displacement (s) is given as 73 m. With these values, we can find the value of acceleration (a).
0^2 = u^2 + 2*a*73
Simplifying the equation, we have:
0 = u^2 + 146a
Now, we have two equations:
v = u + at
0 = u^2 + 146a
To solve these equations, we need to determine the unknown variables u, v, and a.
Now, let's consider the unpowered flight phase. The rocket will reach its maximum height and then fall back to the ground. By conservation of energy, the maximum height will be gained when the rocket's initial kinetic energy (at burnout) is equal to its potential energy at maximum height.
The initial kinetic energy is given by:
KE = (1/2)mv^2
Where m is the mass of the rocket and v is the velocity at burnout.
The potential energy at maximum height is given by:
PE = mgh
Where h is the maximum height reached by the rocket and g is the acceleration due to gravity.
In this case, the velocity at burnout (v) is given as and the maximum height (h) is 73 m.
By equating the initial kinetic energy and potential energy, we have:
(1/2)mv^2 = mgh
Simplifying the equation, we have:
v^2 = 2gh
Now, we have an equation for v^2, which we can use to find the speed upon impact.
By substituting this equation into the existing equation for v^2:
0 = u^2 + 146a
u^2 + 146a = 2gh
We can now solve these equations simultaneously to find the values of u, a, and v.