An arrow is shot horizontally from a 50 m cliff with a velocity of 15 m/s. As the arrow travels in its parabolic path, its new height some time later, is 24 m. Calculate the instantaneous velocity of the moment it reaches is new height.
V^2 = Vo^2 + 2g*d.
V^2 = 0 + 19.6*(50-24) = 509.6
V = 22.6 m/s.
To calculate the instantaneous velocity of the arrow when it reaches its new height, we need to use the principles of projectile motion.
Let's break down the problem into two parts - the horizontal motion and the vertical motion of the arrow.
1. Horizontal Motion:
Since the arrow is shot horizontally, its velocity in the horizontal direction remains constant throughout its flight. In this case, the initial horizontal velocity of the arrow is given as 15 m/s.
2. Vertical Motion:
The arrow is subjected to gravity, causing it to follow a parabolic path. To determine the instantaneous vertical velocity when it reaches a new height, we can make use of the following equation of motion:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
In this case, we want to find the final vertical velocity. Since the arrow is at the peak of its trajectory when it reaches the new height, its final vertical velocity will be 0 m/s.
Now, let's calculate the time taken for the arrow to reach the new height.
We can use the equation for vertical displacement:
s = ut + (1/2)at^2
where:
s = vertical displacement
u = initial vertical velocity (which we can assume is 0 m/s since the arrow is shot horizontally)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation, we have:
t = √(2s / a)
Plugging in the given values, we have:
t = √(2 * 24 / -9.8)
t ≈ 2.77 seconds
Now that we have the time taken for the arrow to reach the new height, we can find the horizontal distance traveled using the equation:
d = v * t
where:
d = horizontal distance
v = horizontal velocity (which we know to be 15 m/s)
t = time taken (2.77 seconds)
Plugging in the values, we have:
d = 15 * 2.77
d ≈ 41.55 meters
Therefore, the horizontal distance traveled by the arrow when it reaches the new height is approximately 41.55 meters.
Finally, to find the instantaneous vertical velocity when the arrow reaches its new height, we use the equation:
v = u + at
where:
v = final vertical velocity (which we want to find)
u = initial vertical velocity (0 m/s)
a = acceleration due to gravity (-9.8 m/s^2)
t = time taken (2.77 seconds)
Plugging in the values, we have:
v = 0 - 9.8 * 2.77
v ≈ -27.07 m/s
Therefore, the instantaneous vertical velocity of the arrow when it reaches its new height is approximately -27.07 m/s.