A coin is dropped from a hot-air balloon that is 325 m above the ground and rising at 12.5 m/s upward. For this problem use a coordinate system in which up is positive.
Find the maximum height reached for the coin in meters
To find the maximum height reached by the coin, we need to first determine how long it takes for the coin to reach its peak height.
When the coin is at its maximum height, its vertical velocity will be zero (since it changes direction from upward to downward). We can calculate the time it takes for the coin to reach this point using the equation of motion:
vf = vi + at,
where
vf is the final velocity,
vi is the initial velocity,
a is the acceleration, and
t is the time.
In this case:
vi = 12.5 m/s (upward, since up is positive),
a = -9.8 m/s² (negative because gravity acts downward),
vf = 0 m/s (at the maximum height).
Rearranging the equation to solve for t gives us:
t = (vf - vi) / a.
Substituting the given values:
t = (0 - 12.5) / -9.8 = 1.28 s.
Therefore, it takes approximately 1.28 seconds for the coin to reach its maximum height.
Now, to find the maximum height reached by the coin, we can use the kinematic equation:
Δy = vi * t + (1/2) * a * t²,
where
Δy is the change in height,
vi is the initial velocity,
a is the acceleration, and
t is the time.
Substituting the values:
Δy = 12.5 * 1.28 + (1/2) * (-9.8) * (1.28)²
= 16.0 + (-7.8) * 1.6384
= 16.0 - 12.748 = 3.252 m.
Therefore, the maximum height reached by the coin is approximately 3.252 meters.
To find the maximum height reached by the coin, we need to consider two factors: the initial height of the balloon and its vertical velocity.
The initial height of the balloon is given as 325 m above the ground, which means the coin is also at this height when it is dropped.
Next, let's consider the vertical velocity of the balloon. It is rising at a speed of 12.5 m/s upward. Since the coin is dropped from the balloon, its velocity will be influenced by the balloon's velocity.
As the coin is dropped, it will initially have a velocity of 0 m/s relative to the balloon. However, since the balloon is rising at 12.5 m/s upward, the coin will also have an initial upward velocity of 12.5 m/s.
To find the maximum height reached by the coin, we can use the kinematic equation:
v_f^2 = v_i^2 + 2aΔy
where:
v_f = final velocity (which is 0 m/s as the coin reaches its maximum height)
v_i = initial velocity (which is 12.5 m/s upward)
a = acceleration (which is the acceleration due to gravity, approximately 9.8 m/s^2)
Δy = change in height (which is the maximum height reached by the coin)
Rearranging the equation, we get:
0 = (12.5)^2 + 2(-9.8)Δy
0 = 156.25 - 19.6Δy
19.6Δy = 156.25
Dividing both sides by 19.6, we get:
Δy = 156.25 / 19.6
Δy ≈ 7.98 m
Therefore, the maximum height reached by the coin is approximately 7.98 meters above the initial height of 325 meters.