An arrow is fired vertically up lands 8 seconds later.Find:
[ A] Its maximum height.
[B] Its initial speed.
It takes the arrow 4 seconds to reach the top and 4 seconds to reach the bottom. Solve for height:
d=1/2at^2=1/2(9.8m/s^2)*(4s)^2=78.4m
Initial velocity=time*deceleration=4s*9.8m/s^2=39.2m
Initial velocity=time*deceleration=4s*9.8m/s^2=39.2m/s
[A] Its maximum height is the place where it stops going up and starts coming back down. In other words, it's the point where it realizes it's reached its peak and can't go any higher. It's like when you're reaching for the last slice of pizza and you realize someone else got to it first - it's a moment of disappointment. But hey, at least the arrow gets to enjoy the view from up there for a little while!
[B] Now, let's talk about the initial speed of the arrow. Imagine this: the arrow wants to take off into the wild blue yonder, so it tries to gather as much speed as possible before leaving the ground. It's like when you're running late for work and you're trying to catch the bus. You sprint, your heart is pounding, your legs are burning, and you hope that you'll make it in time. Well, that's exactly what the arrow does. It gathers all the strength it has and propels itself upwards with an initial speed that can only be described as "fast and furious."
To find the maximum height and initial speed of the arrow, we can use the equations of motion.
Let's break down the information given:
- Time taken to reach the maximum height: 8 seconds.
- The arrow is fired vertically upwards, implying that its initial velocity is in the upward direction.
- We need to find the maximum height (H) reached and the initial speed (u) of the arrow.
Step 1: Determine the time taken to reach the maximum height (t₁):
Since the arrow lands 8 seconds later, it takes half of the total time to reach the maximum height. Therefore, t₁ = 8 seconds ÷ 2 = 4 seconds.
Step 2: Use the equation for displacement to find the maximum height (H):
We can use the equation:
H = u * t₁ + (1/2) * g * t₁²
Where:
H = Maximum height
u = Initial speed (upward)
t₁ = Time taken to reach the maximum height
g = Acceleration due to gravity (-9.8 m/s², assuming downward direction is negative)
Plugging in the values:
H = u * 4 + (1/2) * (-9.8) * (4)²
H = 4u - 19.6
Step 3: Find the initial speed (u):
We can use the equation of motion for vertical motion:
v = u + g * t
Where:
v = Final velocity (at maximum height, the final velocity is 0)
u = Initial speed (upward)
g = Acceleration due to gravity
t = Time taken to reach the maximum height
Plugging in the values:
0 = u + (-9.8) * 4
0 = u - 39.2
u = 39.2 m/s
So, the answers are:
[A] The maximum height reached by the arrow is 4u - 19.6.
[B] The initial speed of the arrow is 39.2 m/s.
To find the answers, we need to understand the motion of the arrow and apply the equations of motion.
Let's break down the problem into two parts:
Part A: Finding the maximum height:
The arrow is fired vertically up and lands 8 seconds later. When the arrow reaches its maximum height, its vertical velocity becomes zero.
We can use the equation for vertical velocity:
v = u + gt
where:
v = final vertical velocity (0 m/s at maximum height)
u = initial vertical velocity (what we want to find)
g = acceleration due to gravity (-9.8 m/s^2, assuming no air resistance)
t = time (unknown)
By substituting the given values:
0 = u - 9.8 * t
Since the arrow is fired vertically and lands 8 seconds later, the time taken to reach the maximum height is half the total flight time (4 seconds):
0 = u - 9.8 * 4
Now, we can solve for u, the initial vertical velocity.
Part B: Finding the initial speed:
To find the initial speed, we need to consider that the initial vertical velocity is equal to the final vertical velocity when the arrow lands. This is because the vertical velocity is affected by gravity, but the horizontal velocity remains constant.
We can use the equation:
v = u + gt
where:
v = final vertical velocity (0 m/s at landing)
u = initial vertical velocity (what we want to find)
g = acceleration due to gravity (-9.8 m/s^2, assuming no air resistance)
t = total flight time (8 seconds)
By substituting the given values:
0 = u - 9.8 * 8
Now, we can solve for u, the initial vertical velocity, which will also be equal to the initial speed.
Once we have the initial speed, we can use it to find the maximum height using the equation for displacement in vertical motion:
s = ut + (1/2)gt^2
where:
s = displacement (maximum height)
u = initial vertical velocity (already found)
t = time to reach the maximum height (4 seconds)
By substituting the values, we can calculate the maximum height.
So, by solving these equations, we can find both the maximum height and the initial speed of the arrow.