An aeroplane flies from a town p on(lat.40N,38E)to another town Q on (Lat.40N,22W).It later flies to a third town T on(Lat.28N,22W). Calculate the average speed at which the aeroplane will fly from P to T via Q, If the journey takes 12 hours. Take R to be 6400 and py to be 22/7.
Nice one
To calculate the average speed of the airplane from P to T via Q, we need to determine the distance traveled and divide it by the total time taken.
Step 1: Calculate the distance between P and Q
First, we need to find the longitudinal distance between P and Q. Since both towns have the same latitude (40N), we can use the following formula to calculate the longitudinal distance:
Longitudinal Distance = R * cos(latitude) * change in longitude
Given:
R = 6400
Latitude = 40N
Change in longitude = 38E - 22W = 38 + 22 = 60 degrees
Using the formula, we can calculate the longitudinal distance between P and Q:
Longitudinal Distance (P to Q) = 6400 * cos(40) * 60
Step 2: Calculate the distance between Q and T
Next, we need to find the longitudinal distance between Q and T. Again, since both towns have the same latitude (28N), we can use the same formula as above with the change in longitude:
Change in longitude = 22W - 22W = 0 degrees
Using the formula, we can calculate the longitudinal distance between Q and T:
Longitudinal Distance (Q to T) = 6400 * cos(28) * 0
Step 3: Calculate the total distance traveled
To find the total distance traveled, we need to sum the distance traveled from P to Q and the distance traveled from Q to T:
Total Distance Traveled = Longitudinal Distance (P to Q) + Longitudinal Distance (Q to T)
Step 4: Calculate the average speed
Finally, to find the average speed, we divide the total distance traveled by the total time taken:
Average Speed = Total Distance Traveled / Time Taken
Given:
Time Taken = 12 hours
Now, plug in the calculated values and solve for the average speed.
To calculate the average speed of the airplane from town P to T via Q, we need to find the distances between the towns and the time it takes for each leg of the journey.
First, let's calculate the distance between town P and Q. The longitude of both towns is given, but we need to calculate the difference in latitude:
Difference in latitude = Absolute value of (Latitude of Q - Latitude of P)
= Absolute value of (40N - 40N)
= 0
The distance between P and Q is given by the formula d = R * Δθ, where R is the radius of the Earth and Δθ is the difference in longitude:
d(PQ) = R * Δθ(PQ)
= 6400 * (38E - 22W)
To simplify the calculations, let's convert the longitudes to the same direction (either east or west):
d(PQ) = 6400 * (38E + 22E) // Since 22W = -22E
= 6400 * 60E
Now, let's calculate the distance between town Q and T. Again, we need to find the difference in latitude:
Difference in latitude = Absolute value of (Latitude of T - Latitude of Q)
= Absolute value of (28N - 40N)
= 12N
The distance between Q and T is given by the formula:
d(QT) = R * Δθ(QT)
= 6400 * (22W - 0E)
Converting the longitudes:
d(QT) = 6400 * (-22E)
= -6400 * 22E // Distance can be negative, indicating the direction (west)
Now that we have the distances for both legs of the journey, let's calculate the total distance:
Total Distance = d(PQ) + d(QT)
Now, to calculate the average speed, we divide the total distance by the total time (12 hours):
Average Speed = Total Distance / Total Time
Finally, substitute the values and calculate the average speed.
Given
R=6400 km
arc=Rθ
The first leg is equi-latitude at 40N with a meridian difference of θx=(38-22)=16°
Distance travelled
=Rθx cos(40)
=6400*(16*π/180)cos(40)
=1369.09 km
The second leg is along a meridian with a latitude difference of
θy(40-28)=12°.
Distance travelled
=Rθy
=6400*(12*π/180)
= 1340.41 km
Total distance = 1369.09+1340.41
= 2709.50 km
Speed=total distance / total time
=2709.50/12
=225.79 km/h