Francesca dangles her watch from a thin piece of string while the jetliner she is in accelerates for takeoff, which takes about 11 s .
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Part A
Estimate the takeoff speed of the aircraft if the string makes an angle of 25∘ with respect to the vertical.
the acceleration (j) of the jet and gravitational acceleration (g) are two sides of the triangle
j / g = tan(25º)
... j = 9.8 m/s^2 * tan(25º)
multiplying j by 11 s will give the takeoff speed
Well, Francesca must really be taking the term "time flies when you're having fun" to a whole new level! But let's get down to business and solve this problem.
We can use a bit of trigonometry here. The angle of 25 degrees with respect to the vertical gives us the angle between the string and the horizontal direction of the jetliner's acceleration.
Now, imagine if Francesca's watch was a superhero cape and it could fly straight out, parallel to the ground. That would be a sight to see! In this imaginary scenario, the angle between the string and the horizontal direction of the jetliner's acceleration would be zero degrees.
So, the difference between the angle the string makes (25 degrees) and the angle it would make if the watch was a superhero cape (0 degrees) gives us the angle of acceleration. This angle of acceleration will help us find the takeoff speed of the aircraft.
Now, if only the watch could tell us the time it took to answer this question. It'd probably say, "Hold on, I need to fly a little longer to give you the answer." But fortunately, we already know the time it took for the takeoff, which is 11 seconds.
To find the takeoff speed of the aircraft, we can use the equation:
takeoff speed = (acceleration due to gravity) * (time taken / sin(angle of acceleration))
However, we need to be careful with the units here. The acceleration due to gravity is usually denoted as "g" and is approximately 9.8 m/s^2. The time taken should also be in seconds. So let's plug in the values:
takeoff speed = (9.8 m/s^2) * (11 s / sin(25 degrees))
Now let's calculate this value and hope that Francesca's watch doesn't get too dizzy while swinging from that string!
To estimate the takeoff speed of the aircraft, we can use the concept of centripetal force. The tension in the string provides the centripetal force, and we can relate it to the gravitational force and the angle made by the string with respect to the vertical.
Let's assume the mass of Francesca's watch is negligible compared to the mass of the aircraft. Therefore, the force providing the centripetal force is the tension in the string.
The gravitational force acting on the watch can be given by:
F_gravity = m * g
where m is the mass of the watch and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The tension in the string can be related to the gravitational force and the angle made by the string with respect to the vertical:
Tension = F_gravity / cos(angle)
where the angle is given as 25 degrees.
Since the tension in the string provides the centripetal force, we can equate this tension to the centripetal force:
Tension = m * v^2 / r
where v is the takeoff speed of the aircraft and r is the length of the string.
By setting the two expressions for tension equal to each other, we can solve for the takeoff speed:
F_gravity / cos(angle) = m * v^2 / r
Rearranging the equation to solve for v:
v = sqrt((F_gravity * r) / (m * cos(angle)))
Substituting the known values:
g ≈ 9.8 m/s^2
angle ≈ 25 degrees
r = length of the string (unknown)
The value of the length of the string is not given, so we cannot estimate the takeoff speed of the aircraft without that information.
To estimate the takeoff speed of the aircraft, we can use basic trigonometry and the concept of centripetal acceleration.
First, let's understand the physical setup. Francesca dangles her watch from a thin piece of string, and the string makes an angle of 25° with respect to the vertical. This means that the string is forming an angle of 65° with respect to the horizontal (complementary to 90° - 25° = 65°).
When the aircraft accelerates for takeoff, the watch and the string experience a centripetal force due to the circular motion caused by the acceleration. The centripetal force is provided by the tension in the string.
Now, let's apply trigonometry to find the vertical component of the tension force. Since the string is at an angle of 25° with respect to the vertical, the vertical component of the tension force is given by:
Vertical component of tension force = Tension force * sin(25°)
Similarly, the horizontal component of the tension force is given by:
Horizontal component of tension force = Tension force * cos(25°)
Since there is no acceleration in the vertical direction (assuming negligible effects from gravity and air resistance), the vertical component of the tension force must balance the weight of the watch:
Vertical component of tension force = Weight of the watch
Assuming the watch has a mass of m, the weight of the watch is given by:
Weight of the watch = m * g
where g is the acceleration due to gravity.
Since we are interested in the takeoff speed of the aircraft, we need to consider the horizontal component of the tension force, which is responsible for the centripetal acceleration. The centripetal force can be calculated as:
Centripetal force = (mass of the watch * velocity^2) / radius
Here, the radius represents the distance from the point where the string is attached to the center of the circular path formed by the watch. However, the radius is not given in the problem statement, so we need more information to calculate the takeoff speed accurately.
To estimate the takeoff speed, we could make some assumptions. One possible assumption is that the string is relatively short, so we can approximate the radius to be the length of the string. If we have the length of the string (l), we can calculate the approximate takeoff speed (v) using the following equation:
Horizontal component of tension force = (m * v^2) / l
From this equation, we can rearrange it to solve for v:
v = sqrt((Horizontal component of tension force * l) / m)
Finally, to get a numerical estimation, we would need to know the values of the mass of the watch, the length of the string, and the horizontal component of the tension force. Without these specific values, we cannot provide an exact answer.