Question

You are part of a searchand- rescue mission that has been called out to look for a lost explorer. You’ve found the missing explorer, but you're separated from him by a 200-m-high cliff and a 30-m-wide raging river. To save his life, you need to get a 5.4 kg package of emergency supplies across the river. Unfortunately, you can't throw the package hard enough to make it across. Fortunately, you happen to have a 1.0 kg rocket intended for launching flares. Improvising quickly, you attach a sharpened stick to the front of the rocket, so that it will impale itself into the package of supplies, then fire the rocket at ground level toward the supplies.

What minimum speed must the rocket have just before impact in order to save the explorer’s life?

Answer 1

To save the explorer's life, the rocket must have a minimum speed just before impact of approximately 145.8 m/s.

This speed allows the rocket to cover the horizontal distance and reach the necessary height to overcome gravitational pull.

Step-by-step explanation:

To calculate this speed, we can use the principle of conservation of energy.

At the starting point, the rocket has a potential energy (mgh) and a kinetic energy (0.5mv^2).

At the highest point, the potential energy becomes zero, and the kinetic energy is the maximum.

Calculate the potential energy at the starting point:

PE = mgh = (5.4 kg) * (9.8 m/s^2) * (200 m)

= 10584 J

Set the potential energy equal to the maximum kinetic energy:

PE = 0.5mv^2

Rearrange the equation to solve for v:

v = sqrt((2PE) / m)

Substitute the known values:

v = sqrt((2 * 10584 J) / (1.0 kg + 5.4 kg))

Calculate the speed:

v ≈ 145.8 m/s