10. A military jet flies horizontally with a velocity of 450 m/s and 20,000 m above the ground. When the jet is straight above an artillery gun a shell is fired. Assuming the shell hits the jet.
a. Calculate the horizontal component of the initial velocity of the shell.
b. Calculate the vertical component of the initial velocity of the shell.
To solve this problem, we can use the concept of projectile motion.
a. To calculate the horizontal component of the initial velocity of the shell, we need to consider that the jet is flying horizontally. Therefore, the horizontal component of the jet's velocity will be the same as the horizontal velocity of the shell.
Horizontal component of initial velocity of the shell = Horizontal velocity of the jet
= 450 m/s
b. To calculate the vertical component of the initial velocity of the shell, we can use the equation of motion for vertical motion under gravity:
h = ut + (1/2)gt^2
where:
h = vertical displacement (20,000 m)
u = initial vertical velocity (vertical component of the shell's velocity)
g = acceleration due to gravity (9.8 m/s^2)
t = time of flight (unknown)
Since the shell is fired when the jet is straight above the artillery gun, and assuming the shell hits the jet, the time of flight for the shell will be the same as the time taken for the jet to fall 20,000 m vertically:
20,000 = (1/2)gt^2
t^2 = (20,000 * 2) / g
t^2 = 40,000 / 9.8
t^2 ≈ 4081.6327
t ≈ √4081.6327
t ≈ 63.874 s (approx.)
Now we can substitute the value of t into the equation of motion for vertical motion under gravity to find the initial vertical velocity:
h = ut + (1/2)gt^2
20,000 = u * 63.874 + (1/2)(9.8)(63.874^2)
20,000 - (1/2)(9.8)(63.874^2) = u * 63.874
u ≈ (20,000 - 31,039.802) / 63.874
u ≈ -11,039.802 / 63.874
u ≈ -172.547 m/s (approx.)
Therefore, the vertical component of the initial velocity of the shell is approximately -172.547 m/s. Note that the negative sign indicates the velocity is directed downwards.
To find the horizontal and vertical components of the initial velocity of the shell, we can use the information given and apply the principles of projectile motion.
First, let's break the initial velocity of the shell into its horizontal and vertical components.
a. To calculate the horizontal component of the initial velocity of the shell:
Since the military jet is flying horizontally, the vertical component of its velocity is zero. Therefore, the horizontal component of the shell's initial velocity will also be zero.
Horizontal component of initial velocity of the shell (Vx) = 0 m/s
b. To calculate the vertical component of the initial velocity of the shell:
We know that the jet is flying 20,000 m above the ground. Let's consider the time it takes for the shell to reach the jet from the ground. During this time, the jet would have moved horizontally at a constant velocity.
The time taken (t) for the shell to reach the jet can be calculated using the formula:
time = distance / velocity
t = 20,000 m / 450 m/s
t ≈ 44.44 s (rounded to two decimal places)
Now, we can calculate the vertical component of the initial velocity of the shell using the time taken:
The vertical displacement of the shell is equal to the vertical displacement of the jet, which is the height at which it is flying. Since the shell hits the jet when it is directly above the artillery gun, the vertical displacement is 20,000 m.
The vertical component of the initial velocity (Vy) can be calculated using the formula:
displacement = initial velocity × time - (1/2) × acceleration × time^2
Considering the initial velocity as Vy, the acceleration due to gravity as -9.8 m/s^2 (taking downwards as positive), and the time as 44.44 s:
20,000 m = Vy × 44.44 s - (1/2) × (-9.8 m/s^2) × (44.44 s)^2
Solving this equation will give the vertical component of the initial velocity (Vy).