Airplane Velocity 210 We are not given a specific amount of time, but we do know that time = distance ÷ rate Without using the calculator, Using the quadratic equation calculator Entering Distance 1 = 750 and Distance 2 = 850 will also work, Clicking "Calculate" we see the answer is: Without using the calculator: Clicking "Calculate" we see the answer is: plane velocity = 680 boat velocity = 10 miles per hour, Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) Since this is a round trip, the distance is the same so: Wind Velocity 12 wind velocity = 7.5 miles per hour We are not given a specific amount of time, but we do know that time = distance ÷ rate Wind (or Current) Velocity 5 725 ÷ (210 + wind velocity) = 675 ÷ (210 - wind velocity) We'll also abbreviate plane velocity and wind velocity as pv and wv. Going downstream, the time would be: (850 × -12) + 850 pv = (750 × 12) + 750 pv 4*plane vel + 160 = 4.5 * plane vel - 180 wind velocity = 40 time2 = 30 ÷ (bv + 5) Distance 2400 What is the boat velocity? Since it's a round trip, the distance is the same so: Using the calculator, we click "D" then enter F) Solving for Boat Velocity when given Current Velocity, Distance and Time. (10,200 + 9,000) ÷ 100 = plane velocity -6* wv + 2,640 = 2,200 +5*wv Distance 1 850 Clicking "Calculate" we see the answer is: 152,250 - 141,750 = 1,400 wind velocity What is the airplane velocity? We'll also abbreviate plane velocity and wind velocity as pv and wv. Clicking "Calculate" we see the answer is: Airplane (or Boat) Velocity 10 We know that time = distance ÷ velocity. Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. Example: When using this method, make sure distance1 is greater than distance2. D) Solving for Wind Velocity given Airplane Velocity and Distance. You take a round-trip on an airplane, that has a velocity in still air of 440 mph. Insert the numbers from the problem into this equation: Insert the numbers from the problem into this equation: 100 pv = 19,200 The typical problem will have some object, a boat or plane for example, which has a known velocity through some medium, air or water, which is itself in motion at a known speed. We'll also abbreviate plane velocity and wind velocity as pv and wv. (entering Time 1 = 6 Time 2 = 5 will also work) (entering Time 1 = 2.75 Time 2 = 3 will also work) Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. boat velocity = 10 miles per hour, Since it's a round trip, the distance is the same so: Example: Wind Velocity 7.5 (10,200 + 9,000) ÷ 100 = plane velocity Time 8 Insert the numbers from the problem into this equation: Without using the calculator, Distance 30 The shorter method: time1 = 30 ÷ (bv - 5) Airplane Velocity 440 In problem 1 and 2 the length of the trip is given, as well as spent time moving in each direction. Insert the numbers from the problem into this equation: Otherwise scroll to "the shorter method". If you want the complete explanation (the solution that algebra teachers like), then just keep reading. -6* wv + 2,640 = 2,200 +5*wv E) Solving for Plane Velocity given Wind Velocity and Distance. wind velocity = 7.5 miles per hour Example: time2 = 30 ÷ (bv + 5) Airplane (or Boat) Velocity 10 Multiplying both sides by (bv -5) 8bv² -60bv -200 = 0 For example, wind blowing from north to south might increase or decrease in ⦠[(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity Insert the numbers from the problem into this equation: Collecting the terms Distance (out) = (440 - wind velocity) * 6 Clicking "Calculate" we see the answers are: Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) time2 = 30 ÷ (bv + 5) Multiplying both sides by (bv +5) Airplane Velocity 210 We'll also abbreviate plane velocity and wind velocity as pv and wv. (10,200 + 9,000) ÷ 100 = plane velocity boat velocity = 10 miles per hour, Time 8 F) Solving for Boat Velocity when given Current Velocity, Distance and Time. (10,200 + 9,000) ÷ 100 = plane velocity [(850 × 12) + (750 × 12)] ÷ (850 -750) = plane velocity The shorter method: Collecting the terms boat velocity = 10 miles per hour, Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) time2 = 30 ÷ (bv + 5) Angle Calculation 0.2 11.3q 650 130 tan 1. plane velocity = 192 Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. time1 = 30 ÷ (bv - 5) (10,200 + 9,000) ÷ 100 = plane velocity We are not given a specific amount of time, but we do know that time = distance ÷ rate Time 2 6 Using the quadratic equation calculator 100 pv = 19,200 Using the calculator, we click "F" then enter Using the calculator, we click "E" then enter We are not given a specific amount of time, but we do know that time = distance ÷ rate (plane vel + 40) * 4 = (plane vel - 40) * 4.5 Clicking "Calculate" we see the answer is: Going downstream, the time would be: Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) The wind velocity is 12 miles per hour. (725 × 210) - (675 × 210) = (725 × wind velocity) + (675 × wind velocity) (entering Time 1 = 6 Time 2 = 5 will also work) Clicking "Calculate" we see the answer is: [ 30 ÷ (bv - 5) ] + [ 30 ÷ (bv + 5) ] = 8 boat velocity = 10 miles per hour, Using the calculator, we click "C" then enter When using this method, make sure distance1 is greater than distance2. A plane flies for 850 miles with the wind and in the same amount of time, it flies 750 miles against the wind. Clicking "Calculate" we see the answers are: There are 2 ways to do this. plane velocity = 192 -10,200 + 850 pv = 9,000 + 750 pv Airplane Velocity 192 Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) (850 ÷ 750) = [(plane velocity + 12) ÷ (plane velocity - 12)] Otherwise scroll to "the shorter method". This gives a value for the angle θ as the difference in wind direction and bearing. Distance 2 750 Going upstream, the time would be: The wind velocity is 12 miles per hour. Clicking "Calculate" we see the answer is: What is the airplane velocity? Time 8 Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. Going upstream, the time would be: The wind velocity is 12 miles per hour. It flies for 725 miles with the wind and in the same amount of time, it flies 675 miles against the wind. Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. time1 = 30 ÷ (bv - 5) Distance = 2,880, C) Solving for Distance and Wind Velocity given airplane velocity and time. We'll also abbreviate plane velocity and wind velocity as pv and wv. Without using the calculator Distance 2 750 wind velocity = 40, Putting this value into this equation: 30bv -150 = 8bv² -70bv +40bv -350 Distance 2 750 Using the calculator, we click "E" then enter Time 8 Time 8 Collecting the terms Without using the calculator, Using the calculator, we click "F" then enter Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. time2 = 30 ÷ (bv + 5) Airplane Velocity 210 We are not given a specific amount of time, but we do know that time = distance ÷ rate Time 1 3 Clicking "Calculate" we see the answer is: Collecting the terms There are 2 ways to do this. 725 ÷ (210 + wind velocity) = 675 ÷ (210 - wind velocity) Using the calculator, we click "E" then enter wind velocity = 7.5 miles per hour Since time1 + time2 = 8 hours, then we can say: We only know the total time, so any equations we write must take that into account. A motorboat (airplane) speed in still water (still air) and the current (wind) speed are unknown. Distance = velocity * time Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. (152,250 -141,750) ÷ 1,400 = wind velocity plane velocity = 192 Distance (out) = (440 - wind velocity) * 6 You take a round-trip on an airplane, that has a velocity in still air of 440 mph. We only know the total time, so any equations we write must take that into account. 10,500 = 1,400 wind velocity Airplane (or Boat) Velocity 10 Distance = (440 - 40) * 6 Clicking "Calculate" we see the answers are: What is the wind velocity? E) Solving for Plane Velocity given Wind Velocity and Distance. Time 1 5 Clicking "Calculate" we see the answer is: Collecting the terms Distance 2400 It flies for 725 miles with the wind and in the same amount of time, it flies 675 miles against the wind. Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) Distance 1 850 340 = .5 * plane velocity Example: We'll also abbreviate plane velocity and wind velocity as pv and wv. We know that time = distance ÷ velocity. (850 × -12) + 850 pv = (750 × 12) + 750 pv Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. It is the responsibility of the user to ensure that the correct values have been selected at all times and to cross-reference the results obtained with a second source. [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -70 Airplane (or Boat) Velocity 10 [ 30 ÷ (bv - 5) ] + [ 30 ÷ (bv + 5) ] = 8 time2 = 30 ÷ (bv + 5) Wind Velocity 12 Using the quadratic equation calculator On the return trip, the airplane flies against a 40 mph wind and takes 4.5 hours to make the trip. -10,200 + 850 pv = 9,000 + 750 pv 440 = 11*wv We only know the total time, so any equations we write must take that into account. Multiplying both sides by (bv -5) [(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity 8bv² -60bv -200 = 0 Going upstream, the time would be: Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. Most problems involving addition of velocity vectors are quite straight forward. The wind velocity is 12 miles per hour. Wind Velocity 12 In this case (5 hours to cover 800 miles), the ground speed is 800/5 = 160 miles per hour). time2 = 30 ÷ (bv + 5) (850 × -12) + 850 pv = (750 × 12) + 750 pv 152,250 - 141,750 = 1,400 wind velocity Dist (out) = (plane vel + 40) * 4 A plane flies for 850 miles with the wind and in the same amount of time, it flies 750 miles against the wind. Otherwise scroll to "the shorter method". We'll also abbreviate plane velocity and wind velocity as pv and wv. We know that time = distance ÷ velocity. 340 = .5 * plane velocity D) Solving for Wind Velocity given Airplane Velocity and Distance. E) Solving for Plane Velocity given Wind Velocity and Distance. Otherwise scroll to "the shorter method". The shorter method: [(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity We'll also abbreviate plane velocity and wind velocity as pv and wv. Clicking "Calculate" we see the answer is: On your trip out, the plane flies for 6 hours against the wind and on your return trip, it flies for 5 hours with the wind. Airplane Velocity 440 (440 - wind velocity) * 6 = (440 + wind velocity) * 5 8bv² -60bv -200 = 0 We'll also abbreviate plane velocity and wind velocity as pv and wv. Using the calculator, we click "F" then enter Distance = velocity * time Example: plane velocity = 192 The velocity of the current is 5 miles per hour. Going upstream, the time would be: Without using the calculator One thing we do know is that the total time equals 8 hours. Multiplying both sides by (bv -5) Using the calculator, we click "F" then enter Distance = (680 +40) * 4 Entering Distance 1 = 750 and Distance 2 = 850 will also work What is the boat velocity? We are not given a specific amount of time, but we do know that time = distance ÷ rate The vector with magnitude $25~\text{mph}$ in the easterly direction represents the velocity of the wind. Entering Distance 1 = 675 and Distance 2 = 725 will also work (725 × 210) - (725 × wind velocity) = (675 × 210) + (675 × wind velocity) -6* wv + 2,640 = 2,200 +5*wv Each plane flies amidst a wind which blows at 20 mi/hr. (152,250 -141,750) ÷ 1,400 = wind velocity The crosswind calculator can be used to quickly and easily determine the parallel and crosswind components of the wind relative to the runway heading. 30 + [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -40 (850 ÷ 750) = [(plane velocity + 12) ÷ (plane velocity - 12)] Wind Velocity 40, Without using the calculator, Going downstream, the time would be: The example problem uses like 3 seperate sketches before getting to an answer, so ⦠(725 × 210) - (725 × wind velocity) = (675 × 210) + (675 × wind velocity) Without using the calculator, You take a round-trip on an airplane, that has a velocity in still air of 440 mph. Distance 30 Note that changing the magnitude of a vector does not indicate a change in its direction. Distance 30 Using the calculator, we click "E" then enter The velocity of the current is 5 miles per hour. 8bv² -60bv -200 = 0 30 + [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -40 Without using the calculator Wind Velocity 7.5 Since time1 + time2 = 8 hours, then we can say: Distance 1 725 boat velocity = 10 miles per hour. Using the calculator, we click "E" then enter Example: A plane flying against the wind flew 270 miles in 3 hours. plane velocity = 192 Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. Without using the calculator, D) Solving for Wind Velocity given Airplane Velocity and Distance. (440 - wind velocity) * 6 = (440 + wind velocity) * 5 Clicking "Calculate" we see the answer is: There are 2 ways to do this. We'll also abbreviate plane velocity and wind velocity as pv and wv. Wind Velocity 12 On your trip out, the plane flies for 6 hours against the wind and on your return trip, it flies for 5 hours with the wind. F) Solving for Boat Velocity when given Current Velocity, Distance and Time. We'll also abbreviate plane velocity and wind velocity as pv and wv. F) Solving for Boat Velocity when given Current Velocity, Distance and Time. Going downstream, the time would be: Distance 2880 Distance = (400) * 6 30 + [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -40 [(850 × 12) + (750 × 12)] ÷ (850 -750) = plane velocity Distance = 2,400 plane velocity = 192 Example: Since time1 + time2 = 8 hours, then we can say: (725 × 210) - (675 × 210) = (725 × wind velocity) + (675 × wind velocity) (850 ÷ 750) = [(plane velocity + 12) ÷ (plane velocity - 12)] 10,500 ÷ 1,400 = wind velocity We only know the total time, so any equations we write must take that into account. The shorter method: Airplane Velocity 192 Time 8 boat velocity = 10 miles per hour, Going upstream, the time would be: A plane's velocity in still air is 210 miles per hour. When using this method, make sure distance1 is greater than distance2. There are 2 ways to do this. Time 2 6 time1 = 30 ÷ (bv - 5) [(850 × 12) + (750 × 12)] ÷ (850 -750) = plane velocity Going upstream, the time would be: 10,500 = 1,400 wind velocity A 20 mph wind from the west is different from a 20 mph wind from the east. If you want the complete explanation (the solution that algebra teachers like), then just keep reading. Airplane Velocity 192, Without using the calculator Otherwise scroll to "the shorter method". [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -70 Collecting the terms Using the quadratic equation calculator 152,250 - 141,750 = 1,400 wind velocity Wind Velocity 40 Distance 2400 Example: There are 2 ways to do this. One thing we do know is that the total time equals 8 hours. The shorter method: The shorter method: Wind (or Current) Velocity 5 Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. time2 = 30 ÷ (bv + 5) 100 pv = 19,200 Distance 30 [(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity Example: Distance (return) = (440 + wind velocity) * 5 Distance 30 Distance = (plane velocity + wind velocity) * time boat velocity = 10 miles per hour, The shorter method: D) Solving for Wind Velocity given Airplane Velocity and Distance. What is the speed of the airplane in still air and the speed of the wind? Without using the calculator (entering Time 1 = 6 Time 2 = 5 will also work) time2 = 30 ÷ (bv + 5) [ 30 ÷ (bv - 5) ] + [ 30 ÷ (bv + 5) ] = 8 Distance 30 Entering Distance 1 = 675 and Distance 2 = 725 will also work