1.A man in a hot-air balloon drops an apple at a height of 150m. If the balloon is rising at 15 m/s, find the highest point reached by the apple.
The height is 150 + 15t - 4.9t^2
The vertex of a parabola is achieved when x = -b/2a = 15/9.8 = 1.53 seconds
so, at t=1.53 seconds, s = 161.48m
Well, to find the highest point reached by the apple, we have to consider the velocities of both the apple and the balloon. So, in this situation, it's like having a balloon race with an apple. It's like the apple saying, "Hey balloon, I'll get higher than you!"
Anyway, let's calculate! The apple starts at a height of 150m and then is dropped, so it only has its initial velocity due to being dropped from rest (which we assume is zero). On the other hand, the balloon is rising at a speed of 15 m/s. So, the highest point reached by the apple will depend on how long it takes to catch up with the balloon.
To find the time it takes to catch up, we can use the equation: distance = velocity × time. The distance the balloon travels is the height of the drop, which is 150m. The velocity of the apple is zero, and the velocity of the balloon is 15 m/s. Solving for time, we get:
150m = 15m/s × time
Time = 150m / 15m/s = 10 seconds
Therefore, the apple will take 10 seconds to catch up to the highest point reached by the balloon. So, the highest point reached by the apple will be at:
Height = initial height + velocity × time
Height = 150m + (15m/s × 10s)
Height = 150m + 150m = 300m
Therefore, the highest point reached by the apple would be 300m above its initial starting point. I hope that puts a smile on your face as high as the apple!
To find the highest point reached by the apple, we need to consider the vertical motion of the apple.
Let's assume the initial velocity of the apple when it is dropped is zero. The only force acting on the apple is gravity, causing it to accelerate downwards at a rate of 9.8 m/s^2.
First, let's find out the time it takes for the apple to reach its highest point. We can use the formula:
vf = vi + at
where:
vf = final velocity (0 m/s at the highest point)
vi = initial velocity (0 m/s when the apple is dropped)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
0 = 0 + (-9.8)t
0 = -9.8t
t = 0 seconds
This means the apple reaches its highest point instantaneously when it is dropped.
To find the highest point, we can use the formula:
d = vi*t + (1/2)a*t^2
where:
d = displacement or distance covered (highest point)
vi = initial velocity (0 m/s when the apple is dropped)
t = time (0 seconds when the apple reaches its highest point)
a = acceleration due to gravity (-9.8 m/s^2)
d = 0*0 + (1/2)(-9.8)(0)^2
d = 0 meters
Therefore, the highest point reached by the apple is 0 meters.
To find the highest point reached by the apple, we need to determine the time it takes for the apple to reach that point. Once we have the time, we can calculate the maximum height using the equation of motion.
First, let's find the time it takes for the apple to reach its highest point. We know that the balloon is rising at a constant velocity of 15 m/s. Let's assume the initial velocity of the apple is 0 since it is dropped from the balloon. We can use the equation:
v = u + at
where:
v = final velocity,
u = initial velocity,
a = acceleration,
t = time taken.
In this case, the final velocity of the apple when it reaches its highest point would be 0 m/s (as it momentarily stops before falling back down). The initial velocity is also 0 m/s. The acceleration due to gravity (assuming no air resistance) is approximately 9.8 m/s^2.
0 = 0 + (-9.8)t
Simplifying the equation, we have:
-9.8t = 0
Since the coefficient of t is 0, we cannot solve for t using this equation. This means that the apple reaches its highest point instantly when it is dropped from the balloon.
Now, let's calculate the height reached by the apple. Since we know the initial velocity is 0 m/s, we can use the equation:
s = ut + (1/2)at^2
where:
s = distance or height,
u = initial velocity,
a = acceleration,
t = time taken.
In this case, the acceleration is -9.8 m/s^2 (negative due to the direction of the acceleration caused by gravity). The time taken is 0 seconds, as we determined earlier.
s = 0(0) + (1/2)(-9.8)(0)^2
s = 0
Therefore, the highest point reached by the apple is at a height of 0 meters.