Question

Самолёт летит горизонтально по прямой со скоростью 300 м/с на высоте 6 км. В тот момент, когда он пролетает точно над зенитным орудием, установленном на поверхности Земли, с него сбрасывают тяжёлое ядро, а орудие производит выстрел. Скорость вылета снаряда из орудия равна 500 м/с, и снаряд попадает точно в ядро. Пренебрегая сопротивлением воздуха, найдите время полёта снаряда. Ответ выразите в с,

Answer 1

1) По горизонтали ядро и снаряд летят равномерно.

Скорость ядра = 300 м/с.

Скорость снаряда = 500*cosα м/с;

300=500*cosα

cosα=0,6

sin²α=1-cos²α=1-0,36=0,64

sinα=0,8

α  -  угол, под которым вылетел снаряд.

2) по вертикали ядро пролетит

h1=gt²/2

а снаряд h2=500*0,8*t - gt²/2  

h2=Voy*t - gt²/2

Voy=Vo*sinα=500*0,8

h1+h2=6000

gt²/2 + 500*0,8t - gt²/2=6000

400t=6000

t=15 c.

Ответ: 15 с.

Answer 2

Problem Analysis

We are given that a plane is flying horizontally in a straight line at a speed of 300 m/s at an altitude of 6 km. At the moment when the plane passes directly over a zenith cannon installed on the surface of the Earth, a heavy core is dropped from the plane, and the cannon fires a projectile with an exit speed of 500 m/s, which hits the core. We need to find the time of flight of the projectile, neglecting air resistance.

Solution

To find the time of flight of the projectile, we can use the concept of relative motion. Since the plane is flying horizontally, the horizontal component of its velocity does not affect the vertical motion of the projectile. Therefore, we only need to consider the vertical motion of the projectile.

Let's assume that the initial height of the projectile above the zenith cannon is h. The time taken by the projectile to reach the core can be found using the equation of motion for vertical motion:

h = ut + (1/2)gt^2

Where: - h is the initial height of the projectile (6 km = 6000 m) - u is the initial vertical velocity of the projectile (0 m/s, as it starts from rest) - g is the acceleration due to gravity (-9.8 m/s^2, assuming downward direction) - t is the time of flight of the projectile (which we need to find)

Rearranging the equation, we get:

(1/2)gt^2 = h

Simplifying further, we have:

t^2 = (2h)/g

Taking the square root of both sides, we get:

t = sqrt((2h)/g)

Substituting the given values, we can calculate the time of flight of the projectile.

Calculation

Using the given values: - h = 6000 m - g = 9.8 m/s^2

We can calculate the time of flight of the projectile using the formula:

t = sqrt((2h)/g)

Substituting the values, we get:

t = sqrt((2 * 6000) / 9.8) = sqrt(1224.49) ≈ 35.00 s

Therefore, the time of flight of the projectile is approximately 35.00 seconds.

Answer

The time of flight of the projectile is approximately 35.00 seconds.