Superman, who likes physics, dangles his watch from a thin piece of string while he is taking off from ground horizontally at a constant acceleration a. He notices that the string makes an angle of θ with respect to the vertical. After 18 seconds, his speed is v=116m/s. Estimate the angle θ.
To estimate the angle θ at which the string makes with respect to the vertical, we can make use of the concept of centripetal acceleration and the tangent function.
First, let's analyze the situation. Superman is taking off from the ground horizontally, which means his initial velocity in the vertical direction is zero. The watch is suspended from a string, which will form an angle with respect to the vertical due to the acceleration. The force acting on the watch to create this acceleration is the tension in the string.
Using Newton's second law, we know that the net force on an object is equal to the mass of the object multiplied by its acceleration. In this case, the mass of the watch is not given, but that will not affect our estimation of the angle θ.
The net force acting on the watch is the vertical component of tension in the string since the watch is only moving vertically. We can express this force as follows:
F_net = T * sin(θ)
The centripetal force required to maintain circular motion (due to the acceleration) is given by the equation:
F_centripetal = m * a
Since the circular motion is in the vertical direction (the string makes an angle with the vertical), we know:
F_centripetal = m * g
where g is the acceleration due to gravity.
Now, let's equate the two forces by setting them equal to each other:
T * sin(θ) = m * g
Now, we can rearrange the equation to solve for sin(θ):
sin(θ) = (m * g) / T
Since we don't have information about the mass or the tension in the string, we cannot determine the exact value of sin(θ). However, we can still estimate the value of the angle by using the given information about Superman's speed and time.
The distance covered by Superman after 18 seconds can be calculated using the formula:
d = (1/2) * a * t^2
where d is the distance, a is the acceleration, and t is the time.
In this case, the distance is not given, but we can use the speed and time to find it. The equation for speed is:
v = a * t
Rearranging the equation, we get:
a = v / t
Plugging in the values, we find:
a = 116 m/s / 18 s = 6.44 m/s^2
Now, we can substitute this value of acceleration into the equation for distance:
d = (1/2) * 6.44 m/s^2 * (18 s)^2 = 2078.56 m
Next, we can use the distance and the angle θ to find the height h (vertical displacement) by employing the trigonometric relationship:
h = d * sin(θ)
Substituting the known values, we have:
h = 2078.56 m * sin(θ)
Finally, we can estimate the angle θ by rearranging the equation:
θ = arcsin(h / 2078.56 m)
Remember, this is only an estimation since we don't have the exact values for mass, tension, or the height displacement h.