An aircraft shell is fired vertically upward with a muzzle velocity of 1000m/s, calculate the maximum height it can attain. calculate the time taken to reach this height. calculate the instaneous velocity at the end of 20s and 50s. calculate when it height will be 37.5km?interpret your result .neglect air resistance
To calculate the maximum height, time taken to reach the maximum height, instantaneous velocity at 20s and 50s, and the time when the height will be 37.5km, we can use the equations of motion for a vertically launched projectile.
1. Maximum Height:
The maximum height can be calculated using the equation:
h = (v₀² sin²θ) / (2g)
where v₀ is the initial upward velocity (1000 m/s), θ is the angle of launch (90 degrees in this case), and g is the acceleration due to gravity (9.8 m/s²). Since the angle is 90 degrees, the equation simplifies to:
h = (v₀²) / (2g)
Substituting the given values, we have:
h = (1000²) / (2 * 9.8)
= 51020.41 m
= 51.02 km
Therefore, the maximum height attained by the aircraft shell is 51.02 km.
2. Time to Reach Maximum Height:
The time taken to reach the maximum height can be calculated using the equation:
t = v₀ sinθ / g
Since the angle of launch is 90 degrees, the equation simplifies to:
t = v₀ / g
Substituting the given values, we have:
t = 1000 / 9.8
= 102.04 s
Therefore, the time taken to reach the maximum height is 102.04 seconds.
3. Instantaneous Velocity at 20s and 50s:
The instantaneous velocity of the aircraft shell can be calculated using the equation:
v = v₀ - gt
At t = 20s:
v = 1000 - (9.8 * 20)
= 802 m/s
At t = 50s:
v = 1000 - (9.8 * 50)
= 510 m/s
Therefore, the instantaneous velocities at 20s and 50s are 802 m/s and 510 m/s, respectively.
4. Time When Height is 37.5km:
To find the time when the height is 37.5 km, we need to consider the downward motion of the shell after reaching the maximum height. The equation to find the time is:
h = v₀t + (1/2)gt²
Rearranging the equation to isolate t, we have:
t = (√(2gh - v₀²) - v₀) / g
Substituting the given height (37.5 km = 37500 m), initial velocity (1000 m/s), and acceleration due to gravity (9.8 m/s²), we get:
t = (√(2 * 9.8 * 37500 - 1000²) - 1000) / 9.8
= (√(7050000 - 1000000) - 1000) / 9.8
= (√6050000 - 1000) / 9.8
= (2459.62 - 1000) / 9.8
= 1459.62 / 9.8
≈ 149.01 s
Therefore, the time when the height is 37.5 km is approximately 149.01 seconds.
Interpretation:
The maximum height attained by the aircraft shell is 51.02 km and it takes 102.04 seconds to reach this height. The instantaneous velocity at 20s is 802 m/s and at 50s is 510 m/s. Finally, the height of 37.5 km is attained approximately after 149.01 seconds. These calculations assume no air resistance and the shell is fired vertically upward.