A plane is flying north at 175 km/hr. The plane gradually turns to the west, maintaining a constant speed. During the turn the path traces a quarter circle. Once the plane is pointing west, it continues on in a straight line in that direction.
Oh, I see we have a pilot in the making! How can I tickle your funny bone with this question? Well, here's a joke for you:
Why did the plane go to therapy?
Because it had a lot of baggage and needed to find its true direction!
Now, back to your question. Since the plane is flying in a quarter circle, and it maintains a constant speed, we know that its speed is 175 km/hr. Without knowing the radius of the turn, we can't determine the exact distance covered during the turn. But once the plane points west, it continues in a straight line until its destination.
To find the radius of the quarter circle path, we need to use the concept of centripetal acceleration. Let's assume that the $r$ represents the radius of the circular path.
Given:
- Velocity of the plane, $v = 175 \, \text{km/hr}$
- Path traced by the plane, a quarter circle
The centripetal acceleration, $a_c$, is given by the formula:
\[ a_c = \frac{v^2}{r} \]
In this case, the centripetal acceleration is caused by the plane turning towards the west. Once the plane completes the turn and starts moving in a straight line, there will be no centripetal acceleration.
Step 1: Convert the velocity from km/hr to m/s
We need to convert the velocity from kilometers per hour to meters per second.
1 kilometer = 1000 meters
1 hour = 3600 seconds
Therefore, the velocity in m/s is:
\[ v = \frac{175 \, \text{km/hr}}{3.6} \]
Step 2: Calculate the radius of the circular path
Using the formula for the centripetal acceleration, we can solve for the radius.
\[ a_c = \frac{v^2}{r} \]
Step 3: Substitute the given values
Substitute the known values into the equation:
\[ \frac{v^2}{r} = a_c \]
Substituting the value of $v$:
\[ \frac{\left(\frac{175 \, \text{km/hr}}{3.6}\right)^2}{r} = a_c \]
Step 4: Solve for the radius
Rearrange the equation and solve for $r$:
\[ r = \frac{\left(\frac{175 \, \text{km/hr}}{3.6}\right)^2}{a_c} \]
Since we do not have a specific value for the centripetal acceleration, we cannot calculate the radius without additional information.
To analyze the motion of the plane, let's break it down into two parts: the circular path while turning and the straight line motion afterward.
1. Circular Path:
The plane is initially flying north and gradually turns to the west while maintaining a constant speed. During the turn, the path traces a quarter circle. To find the radius of the circular path, we need additional information. Let's assume that the plane takes 10 minutes (0.167 hours) to complete the turn.
To determine the radius of the circular path, we can use the formula for centripetal acceleration:
a = v^2 / r
Given:
v (velocity) = 175 km/hr
t (time) = 0.167 hours
We can find the centripetal acceleration. Rearranging the formula:
a = v^2 / r
r = v^2 / a
The centripetal acceleration can be calculated using the formula:
a = Δv / t
Since the velocity is constant, the change in velocity (Δv) is zero.
a = Δv / t = 0 / 0.167 = 0 m/s^2
Plugging the values into the radius formula, we get:
r = (v^2) / a = (175 km/hr)^2 / 0 m/s^2
To complete the calculation, we need to convert the velocity to m/s:
175 km/hr * (1000 m / 1 km) * (1 hr / 3600 s) = 48.611 m/s
Now we can substitute the values:
r = (48.611 m/s)^2 / 0 m/s^2
Since the denominator is zero, it means that the centripetal acceleration is not applicable. This scenario is not physically possible to sustain a circular path without any acceleration. Therefore, the circular motion is not achievable.
2. Straight Line Motion:
After the plane completes the turn, it heads west in a straight line. Since the circular path is not possible, the plane would continue straight rather than follow a curved path.
In summary, the plane cannot trace a quarter circle while turning because it would require a centripetal force. Instead, it would continue straight in the direction it is heading after the turn.