A toy rocket is launched vertically from ground level (y = 0 m), at time t = 0.0 s. The rocket engine provides constant upward acceleration during the burn phase. At the instant of engine burnout, the rocket has risen to 73 m and acquired a velocity of The rocket continues to rise in unpowered flight, reaches maximum height, and falls back to the ground. The speed of the rocket upon impact on the ground is closest to:

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.