Vectors and Projectiles Legacy Problem #3 Guided Solution
Problem*
The takeoff speed of a military aircraft from an aircraft carrier is approximately 170 mi/hr relative to the air. They acquire this speed through a combination of a catapult system present on the aircraft carrier and the aircraft's jet propulsion system. A common strategy is to head the carrier and the plane into the wind. If a plane is taking off from an aircraft carrier which is moving at 40 mi/hr into a 20 mi/hr headwind, then what speed relative to the deck of the aircraft carrier must it obtain to takeoff?
Audio Guided Solution
This is a problem whose only mathematical skill involves 3rd grade arithmetic. Yet it's an abnormally difficult problem because you need to be able to figure out that the only thing this involves is arithmetic. You have to picture the situation of this military aircraft that wants to take off. And according to the problem, it must be traveling at 170 miles per hour relative to the air. Now if the aircraft was on land and there was no wind, then it would have to go 170 miles per hour relative to the runway. But it's on a runway that's moving and it's moving at 40 miles per hour into the wind. Thus we don't have to go 170 miles per hour relative to the runway because we're on a runway that's actually moving. We only have to go 130 miles per hour relative to that runway. But not only that, we have a wind, not still air, and the wind is moving 20 miles per hour against the plane. And so that's another vector that we can consider here. And so now instead of having to go 130 miles per hour relative to the runway, we only need to go 110 miles per hour relative to the runway. The fact that the aircraft carrier is going 40 miles per hour relative to the water, and then that the air is going in the opposite direction, 20 miles per hour relative to the water, means that we get an extra 60 miles per hour boost out of all this. So we can only need to go 110 miles per hour relative to the aircraft carrier deck.
Solution
110 mi/hr
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(v_\text{ox} = \units{12.4}{\unitfrac{m}{s}}\), \(v_\text{oy} = \units{0.0}{\unitfrac{m}{s}}\), \(d_x = \units{32.7}{m}\), \(d_y = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Vectors and Projectiles at The Physics Classroom Tutorial.