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 13 m/s at an angle of 68.0°
above the horizontal. At the instant when the kit was caught, it is traveling
horizontally, so its vertical speed is zero. What is the vertical height between
the two climbers?
To find the vertical height between the two climbers, we can use the kinematic equations of motion in projectile motion. We know the initial velocity of the kit, the angle at which it was thrown, and that its vertical speed at the instant of catching is zero.
Let's break down the given information:
- Initial velocity (v0) = 13 m/s
- Angle (θ) = 68.0°
- Vertical speed at the instant of catching (vy) = 0 m/s
We need to find the vertical height (Δy).
First, we can find the time taken for the kit to reach the other climber using the vertical component of velocity:
vy = v0 * sin(θ)
0 = 13 * sin(68.0°)
sin(68.0°) = 0
Since sin(68.0°) is not equal to zero, it means that the kit will take some time to reach the other climber. Therefore, we can conclude that the kit is not thrown directly horizontally.
Now, let's find the time of flight (t) of the kit using the vertical component of velocity:
vy = v0 * sin(θ)
0 = 13 * sin(68.0°)
sin(68.0°) = 0
We know that the vertical speed is zero when the kit is caught, which means it will take the same amount of time to reach the highest point and fall back down. Therefore:
t = 2 * time to reach maximum height
Since we need the time to reach maximum height, let's focus on finding that first. To calculate this, we need the vertical component of velocity when the kit reaches the highest point.
In projectile motion, the time to reach maximum height can be found using the formula:
t_max = vy / g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, we can substitute the values and find the time to reach maximum height:
t_max = (13 * sin(68.0°)) / 9.8
Once we have the time to reach the maximum height, we can calculate the total time of flight by multiplying it by 2:
t = 2 * t_max
Now, let's find the vertical height (Δy) traveled by the kit:
Δy = vy * t - (0.5 * g * t^2)
Since vy is 0, the equation simplifies to:
Δy = -0.5 * g * t^2
Substituting the values, we can now calculate the vertical height:
Δy = -0.5 * 9.8 * (2 * (13 * sin(68.0°)) / 9.8)^2
Simplifying the equation, we get:
Δy = -sin(68.0°) * (13)^2
Evaluating this expression, we find the vertical height (Δy).