Given: A rock is thrown upward from the surface of a planet at 24m/s. It reaches a maximum height of 180 meters.
Question: How fast is the rock falling when it lands on the planet?
I am honestly dumbfounded by this problem. I feel that I don't have enough information, I don't even know where to approach. Shouldn't there be a position function?
Any input is appreciated. Thank you
This is a trick question, in a way. You can go through the math of height and speed, using an unknown acceleration (you will find that this fails to have a solution when g=9.8 is used).
Or, you can just note that since momentum is conserved, the speed at launch is exactly the same as the speed at impact: 24 m/s
To solve this problem, you can use the kinematic equations of motion. Although it might seem like you don't have enough information, you can still find the answer by considering the symmetry of the motion.
Here's how you can approach the problem step by step:
1. Recall the kinematic equation for vertical motion:
h = u*t - (1/2)*g*t^2
Where:
- h is the height
- u is the initial velocity
- t is the time
- g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
2. In this case, the initial velocity (u) is 24 m/s and the maximum height (h) is 180 meters. By substituting these values into the equation, you can find the time it takes for the rock to reach its maximum height. Rearrange the equation to solve for time (t):
180 = 24*t - (1/2)*9.8*t^2
This is a quadratic equation, so solve it using the quadratic formula or any other method you prefer.
3. Once you find the time it takes for the rock to reach its maximum height, note that the time it takes to reach the maximum height is equal to the time it takes to fall back down to the surface due to the symmetry of motion. Let's call this time "T".
4. Now, you can determine the final velocity (v) when the rock lands on the planet. Use the equation:
v = u - g*T
Where:
- v is the final velocity
- u is the initial velocity
- T is the time it takes for the rock to reach its maximum height and fall back down
5. Substitute the values you have into the equation:
v = 24 - 9.8*T
6. Finally, calculate the value for v using the value of T you found in step 2.
By following these steps, you should be able to find the velocity at which the rock is falling when it reaches the planet's surface.