A weather balloon carrynig a data measurement(DM) is rising quickly at a constant velocity, U. when its 60m above ground, balloon pops and the dm experiences freefall. It reaches ground 4 seconds after the balloon pops. If DM set hits ground at speed greater than 40 ms-1 it will break.
Explain whether the DM set broke. showing your working clearly.
State whether the dm set would have broke if the weather balloon was moving at a constant velocity of -U if it pops
the height of the DM after the balloon pops is given by
y(t) = 60 + Ut - 4.9t^2
If y(4) = 0, then
60+4U-78.4 = 0
U = 4.6
The speed of the DM is given by
v(t) = U-9.8t
v(4) = U-39.2
So, as long as U > 0.8, the DM will hit the ground with v > -40m/s and will not break.
To determine if the DM set broke, we need to analyze the motion of the DM set before and after the balloon popped.
Let's calculate the time it took for the DM set to reach the ground after the balloon popped:
Given:
Height when the balloon pops, h = 60m
Time taken to reach the ground after the balloon pops, t = 4s
We can use the equation for free fall motion to find the time it took for the DM set to fall from the height h:
h = ut + (1/2)gt^2
Since the balloon was rising at a constant velocity, the initial velocity (u) is equal to the velocity of the balloon, which is U. The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Substituting the values into the equation:
60 = U(4) + (1/2)(9.8)(4^2)
60 = 4U + (1/2)(9.8)(16)
60 = 4U + (4.9)(16)
60 = 4U + 78.4
4U = -18.4
Dividing both sides by 4:
U = -4.6 m/s
The velocity of the weather balloon is found to be -4.6 m/s.
Now, let's analyze whether the DM set broke:
Given:
Threshold speed for the DM set to break, v_threshold = 40 m/s
If the DM set reaches the ground at a speed greater than 40 m/s, it will break.
Since the velocity of the DM set is the same as the velocity at which the balloon was moving before it popped, we need to determine if this velocity exceeds the threshold speed.
If the velocity, U = -4.6 m/s, is less than -40 m/s, then it means the DM set reached the ground at a speed greater than 40 m/s, and therefore it broke.
However, if the velocity, U = -4.6 m/s, is greater than -40 m/s, then it means the DM set did not reach the ground at a speed greater than 40 m/s, and therefore it did not break.
By evaluating the velocity (-4.6 m/s) and comparing it to the threshold speed (-40 m/s), we can determine whether the DM set broke or not.