A rock climber throws a small first aid kit to another climber who is higher up the mountain. The initial velocity of the kit is 23 m/s at an angle of 69° above the horizontal. At the instant when the kit is caught, it is traveling horizontally, so its vertical speed is zero. What is the vertical height between the two climbers?
its initial vertical speed was 23Sin69
its final vertical speed was zero.
Vf^2=Vi^2-2*9.8height
solve for height
Well, well, well, it looks like we have some high-flying climbers here. Let's do some calculations to find that vertical height, shall we?
Now, since the kit is traveling horizontally when it is caught, we know that its vertical speed is zero. So we need to find the time it takes for the kit to reach that point.
First, let's break down the initial velocity into its vertical and horizontal components. The vertical component can be found with some trigonometry: 23 m/s * sin(69°). This will give us the vertical speed at the start.
Now, to find the time it takes for the kit to reach zero vertical speed, we can use the equation v = u + at. Since the final vertical speed is zero, we can rewrite the equation as 0 = vertical start speed + (-9.8 m/s^2) * t. Solve this equation for time (t).
Once you have the time it takes for the kit to reach zero vertical speed, you can use another equation to find the vertical height. The equation is s = ut + (1/2)at^2. Here, s represents the vertical distance traveled, u represents the initial vertical speed, a represents the acceleration due to gravity (-9.8 m/s^2), and t represents the time.
Plug in all the values you have, and voila! You'll find the vertical height between the two climbers. Keep in mind that my calculations might be a bit wobbly, but I hope they give you a good chuckle. Happy climbing!
To find the vertical height between the two climbers, we need to find the time it takes for the first aid kit to reach the higher climber's position.
First, let's break down the initial velocity into its vertical and horizontal components.
The vertical component of the initial velocity can be found by multiplying the initial velocity (23 m/s) by the sine of the angle (69°) above the horizontal:
Vertical component = 23 m/s * sin(69°) = 23 m/s * 0.9397 ≈ 21.59 m/s
Since the kit reaches a point where its vertical speed is zero when it is caught, we know that the time it takes to reach that point is equal to the time it takes to reach the peak of its vertical motion. At the peak, its vertical speed would be zero before it starts moving downward.
We can find the time it takes to reach the peak by using the vertical component of the initial velocity:
Vertical component = (final vertical velocity) - (initial vertical velocity)
0 m/s = (-21.59 m/s) - 10 m/s * t
Simplifying the equation, we have:
-21.59 m/s = -10 m/s * t
Solving for t gives us:
t ≈ 2.16 s
Now, we know it takes approximately 2.16 seconds for the kit to reach the peak of its vertical motion.
To find the vertical height between the two climbers, we can use the time (2.16 s) and the initial vertical velocity (21.59 m/s):
Vertical height = (initial vertical velocity) * time + 0.5 * (acceleration due to gravity) * (time)^2
Plugging in the values, we have:
Vertical height = 21.59 m/s * 2.16 s + 0.5 * (-9.8 m/s^2) * (2.16 s)^2
Simplifying the equation, we get:
Vertical height ≈ 25.81 m
Therefore, the vertical height between the two climbers is approximately 25.81 meters.