The positions of a particle and a thin (treat it as being as thin as a line) rocket of length 0.280 m are specified by means of Cartesian coordinates. At time 0 the particle is at the origin and is moving on a horizontal surface at 23.0 m/s at 51.0°. It has a constant acceleration of 2.43 m/s2 in the +y direction. At time 0 the rocket is at rest and it extends from (−.280 m, 50.0 m) to (0, 50.0 m), but, it has a constant acceleration in the +x direction. What must the acceleration of the rocket be in order for the particle to hit the rocket?
You only need one physics equation to solve this problem: The distance formula.
The formula to determine the distance traveled by a moving object during a given time is:
distance = initial distance + initial velocity x time + 0.5 x acceleration x time2
Can you solve the problem?
All you need to solve the problem is the above formula and the math you should have learned in high school.
Let me give you a hint: The particle will collide with the nose of the rocket when it reaches the height (50.0 m) of the rocket since the rocket is flying level. How far along the x axis is the rocket at that point and how long did it take to get there? What is the acceleration at this time. What if the rocket hit the tail of teh rocket instead. What is the rocket's acceleration now.
Now you know the steps to solve the problem. Now get to it.
Can't do it? Let me make it easier for you by telling you every step of the solution.
1. Decompose the velocity of the particle into its x and y components. You need to know simple trigonometry and algebra to do this. Here's another hint: The sine of an angle equals the opposite side divided by the hypoteneuse. The cosine of an angle equals the adjacent side divided by the hypoteneuse.
2. Now use the distance formula and determine how long it will take for the particle to reach the height of the rocket. You'll need to know how to solve a quadratic equation to do this.
3. Now determine how far along the ground the particle will be when it reaches the height of the rocket.
4. Using the distance formula again determine the acceleration of the rocket if the particle were to strike the nose. This step requires simple algebra to solve.
5. Using the distance formula again determine the acceleration of the rocket if the particle were to strike the tail. This step requires simple algebra to solve.
6. The answer is that the acceleration of the rocket must be between the aceleration of the nose and the acceleration of the tail when the particle strikes the rocket or else it misses.
Ok, I've set-up the problem for you. I've conceptualized the problem for you. I've given you the physics. All you need to do is to use your high school math to solve.
Go ahead give it a whirl. It's still not an easy problem to solve is it? Especially if your algebra and/or trig skills are rusty. Try guessing and checking your way to the solution. I dare you to try.
If you're uncertain how to calculate an arcsine, how distracted do you think you'll be having to shift gears and ponder that while the physics problem is waiting to be solved. Or, maybe you are a real smartie and know how to derive an arcsine from first principles, go ahead work it out. The physics problem will still be waiting when you're done. Gee, I bet that was distracting.
Now we get to the quadratic equation. Remember how to solve one of those off the top of your head? That means you have to derive it yourself or go look it up. The physics problem is getting impatient now. Do you even remember where you were in the solution. What's the next step? What are you trying to solve anyway using the quadratic formula? What do you do with the extra solution you get when you finally solve the quadratic equation? Now all you have to do is use the distance formula to solve for three different variables. To do that you'll need to know how to manipulate algebraic expressions with some skill. Are you confused. Does your brain hurt yet?
This is a simple college level physics problem the likes of which is going to confront every aspiring science and engineering student in their first year of undergrad. It gets a lot tougher than this real quick.
This is one of the first gatekeepers students will meet. Most will not get through the gate. Most will not get through the gate because they did not have the domain knowledge in math to enable them to quickly solve the hundreds of physics problems they'll need to solve in order to acquire the domain knowledge in phsyics in order to pass their Physics course.
Once students have the physics domain knowledge, solving problems like this is simple and requires little cognitive ability. What seems like an insurmountable problem to the novice, requiring enormous cognitive effort, turns out to be a simple problem for the expert. One that can be easily broken down into simple steps that can be solved using one formula and some high school math.
Here's another real world question. Can you determine whether a student scoring at the advanced level of the 11th grade NAEP or any NCLB state exam knows the math necessary to do my real-world physics problem.
Answer: No. There is nothing on either the NAEP or any state exam even remotely approaching the difficulty level of math needed to do the problem. Less than 10% of students perform at the advanced NAEP level.