Two planes fly from toronto to
philadelphia. Plane a flies via pittsburgh whereas passenger on plane B have a direct flight. Pittsburgh is 350km due south of Toronto and 390km due west of Philadelphia. The airspeed of both planes is 400km/h and a steady wind is blowing from the east at 60km/h
a) What direction must the pilot point the plane flying from toronto to pittsburgh?
b) How long will the entire flight take for plane A assuming a 0.50-h layover in pittsburgh?
c) How mich time must the pilot of plane B wait before leaving toronto if she is to arrive in philadelphia at the same time plane A arrives?
(wat i really need is how to draw the diagram for the first question and how to get started for the rest of the questions and i can take it from there)
To answer the first question, you can start by drawing a diagram. Let's use a coordinate system with Toronto at the origin (0,0). Pittsburgh is 350 km due south, so its coordinates are (0, -350). Philadelphia is 390 km due west, so its coordinates are (-390, 0).
Next, draw the vector representing the wind. Since the wind is blowing from the east at 60 km/h, the vector representing the wind will point west with a magnitude of 60 km/h.
To determine the direction that the pilot of Plane A must point the plane, you need to find the resultant vector of the plane's airspeed and the wind velocity. You can do this by vector addition.
The airspeed of Plane A is 400 km/h, and the wind velocity is 60 km/h to the west. Since the wind is blowing against the plane, you subtract the wind velocity vector from the airspeed vector.
To compute the resultant vector, subtract the x-components and y-components separately:
x-component: 400 km/h - 60 km/h = 340 km/h (to the east)
y-component: 0 km/h (there is no wind blowing north or south)
Hence, the pilot of Plane A must point the plane in the east direction.
For the second question, to find the time taken for the entire flight for Plane A, you have to calculate the distance between Toronto and Pittsburgh and add it to the distance between Pittsburgh and Philadelphia.
The distance between Toronto and Pittsburgh is the y-component of the position vector for Pittsburgh, which is 350 km.
The distance between Pittsburgh and Philadelphia is the x-component of the position vector for Philadelphia, which is 390 km.
The total distance traveled is: 350 km + 390 km = 740 km.
To find the time taken, you can divide the total distance by the airspeed of Plane A: 740 km ÷ 400 km/h = 1.85 hours.
Since there is a 0.50-hour layover in Pittsburgh, the entire flight for Plane A will take 1.85 hours + 0.50 hours = 2.35 hours.
For the third question, the time that the pilot of Plane B must wait before leaving Toronto can be determined by calculating the time taken for Plane A to complete its journey.
From the previous calculation, we found that the total flight time for Plane A is 2.35 hours.
To arrive in Philadelphia at the same time as Plane A, Plane B needs to depart Toronto 2.35 hours minus the time taken to travel from Toronto to Pittsburgh.
The time taken to travel from Toronto to Pittsburgh can be found by dividing the distance by the airspeed: 350 km ÷ 400 km/h = 0.875 hours.
Therefore, Plane B needs to wait 2.35 hours - 0.875 hours = 1.475 hours (or approximately 1 hour and 28 minutes) before leaving Toronto in order to arrive in Philadelphia at the same time as Plane A.
To answer these questions, we can start by drawing a diagram. Let's represent Toronto as point T, Pittsburgh as point P, and Philadelphia as point Philly. The wind is blowing from the east (right to left on our diagram), so we'll draw an arrow opposite to the direction of the wind.
a) To figure out the direction the pilot of plane A must point, we need to consider the combined effect of the wind and the plane's airspeed. Since the wind is blowing from the east at 60 km/h and the plane's airspeed is 400 km/h, we can add these vectors together to determine the resultant velocity.
Drawing a line 60 km/h to the left from Toronto (T) will give us the wind vector. Then, connecting this vector to the 400 km/h plane's airspeed vector (drawn from Toronto (T) to Pittsburgh (P)) will give us the resultant velocity vector. The direction of the resultant velocity vector is the direction in which the pilot should point the plane.
b) To calculate the total time for plane A's journey, we can consider the distances and speeds involved. From the information given, we know that Toronto to Pittsburgh is 350 km and Pittsburgh to Philadelphia is 390 km. The airspeed of the plane is 400 km/h.
First, we find the time it takes to travel from Toronto to Pittsburgh. Since we already know the distance and speed, we can use the formula:
time = distance / speed
Let's calculate it:
time for Toronto to Pittsburgh = 350 km / 400 km/h
Next, we find the time it takes to travel from Pittsburgh to Philadelphia. Again, using the formula:
time = distance / speed
time for Pittsburgh to Philadelphia = 390 km / 400 km/h
Finally, if there is a 0.50-hour layover in Pittsburgh, we need to add this layover time to the total time calculated above.
c) To determine the amount of time the pilot of plane B must wait before leaving Toronto in order to arrive in Philadelphia at the same time as plane A, we need to compare the times calculated for plane A's journey.
Since plane A has a layover of 0.50 hours in Pittsburgh, we need to subtract this layover time from the total time of plane A's journey. Let's call this adjusted total time TA.
To find the waiting time for plane B, which has a direct flight, we can subtract the time it takes for plane A to travel from Toronto to Pittsburgh from the adjusted total time TA.
waiting time for plane B = TA - (time for Toronto to Pittsburgh)
Now, let's plug in the values and calculate the answers based on the diagram and the explanations given above.
a) The "airspeed" vector has a magnitude of 400 km/h and a direction in which the pilot points the plane (which is what they are asking you to calculate). The wind velocity vector has magnitude 60 km/h and direction TOWARD the west. The ground velocity vector of the plane going to Piitsburgh has a direction due south. It is the sum of the airspeed and windspeed vectors. You have two sides and one angle of the vector-addition triangle, so can solve for all unknowns.
b) Get the ground speed of the Pittsburhg leg of the trip from the vector triangle opf the last problem. Divide 350 km by that speed for the elapsed time. For the leg of the trip to Philly, the ground speed is just 400 - 60 = 340 km/h. Compute the elapsed time for that leg, and add the first leg elapsed time plus 0.5 hours.
c) Compute the travel time for the direct flight using the same method to get the ground velocity vector. You know its direction already. Compare that to the answer in (b) to get the required waiting time before takeoff.