Question

Из пушки выпустили последовательно два снаряда со скоростью v0 = 216 м/с: первый — под углом ϑ1 = 55° к горизонту, второй — под углом ϑ2 = 16° . Пренебрегая сопротивлением воздуха, найти интервал времени между выстрелами, при котором снаряды столкнутся друг с другом.

Answer 1

Уравнения движения тела, брошенного под углом к горизонту:
$x(t)=v_0t\cos\alpha\\ y(t)=v_0t\sin\alpha-\dfrac{gt^2}2$

Пусть первый снаряд до встречи пролетел время t + τ, второй t. Во время столкновения координаты снарядов равны:
$\begin{cases} (t+\tau)v_0\cos\theta_1=tv_0\cos\theta_2\\ (t+\tau)v_0\sin\theta_1-\dfrac{g(t+\tau)^2}2=tv_0\sin\theta_2-\dfrac{gt^2}2 \end{cases}$

Из первого уравнения
$t=\tau\cdot\dfrac{\cos\theta_1}{\cos\theta_2-\cos\theta_1},\;t+\tau=\tau\cdot\dfrac{\cos\theta_1}{\cos\theta_2-\cos\theta_1}$

Подставляем во второе уравнение.
$v_0\tau\dfrac{\sin\theta_1\cos\theta_2}{\cos\theta_2-\cos\theta_1}-\dfrac{g\tau^2}2\dfrac{\cos^2\theta_2}{(\cos\theta_2-\cos\theta_1)^2}=v_0\tau\dfrac{\sin\theta_2\cos\theta_1}{\cos\theta_2-\cos\theta_1}-\\-\dfrac{g\tau^2}2\dfrac{\cos^2\theta_1}{(\cos\theta_2-\cos\theta_1)^2}$

Переносим квадраты τ в одну часть, остальное в другую и делим на τ.
$\dfrac{g\tau}2\cdot\dfrac{\cos^2\theta_2-\cos^2\theta_1}{(\cos\theta_2-\cos\theta_1)^2}=v_0\dfrac{\sin\theta_1\cos\theta_2-\sin\theta_2\cos\theta_1}{\cos\theta_2-\cos\theta_1}\\ \dfrac{g\tau}{2}\dfrac{\cos\theta_1+\cos\theta_2}{\cos\theta_2-\cos\theta_1}=v_0\dfrac{\sin(\theta_1-\theta_2)}{\cos\theta_2-\cos\theta_1}$

Находим значение τ.
$\tau=\dfrac{2v_0}g\dfrac{\sin(\theta_1-\theta_2)}{\cos\theta_1+\cos\theta_2}$

Подставляем значения:
$\tau=\dfrac{2\cdot216}{9.8}\cdot\dfrac{0.63}{0.57+0.96}\text{ s}=18.2\text{ s}$

Ответ. 18,2 с

Answer 2

Problem Analysis

We are given that two projectiles are fired from a cannon with an initial velocity of v0 = 216 m/s. The first projectile is fired at an angle of ϑ1 = 55° to the horizontal, and the second projectile is fired at an angle of ϑ2 = 16° to the horizontal. We need to find the time interval between the two shots at which the projectiles collide with each other. We can assume that there is no air resistance.

Solution

To solve this problem, we can use the equations of motion for projectile motion. The horizontal and vertical components of the motion are independent of each other.

Let's start by finding the time of flight for each projectile. The time of flight is the time it takes for the projectile to reach the ground. We can use the following equation:

Time of flight (T) = (2 * v0 * sin(ϑ)) / g, where v0 is the initial velocity, ϑ is the angle of projection, and g is the acceleration due to gravity.

For the first projectile: - v0 = 216 m/s - ϑ1 = 55°

Using the equation, we can calculate the time of flight for the first projectile.

T1 = (2 * 216 * sin(55°)) / g Similarly, for the second projectile: - v0 = 216 m/s - ϑ2 = 16°

T2 = (2 * 216 * sin(16°)) / g Now, let's find the horizontal distance covered by each projectile. The horizontal distance is given by:

Horizontal distance (D) = v0 * cos(ϑ) * T, where v0 is the initial velocity, ϑ is the angle of projection, and T is the time of flight.

For the first projectile: - v0 = 216 m/s - ϑ1 = 55° - T1 = calculated above

D1 = 216 * cos(55°) * T1 Similarly, for the second projectile: - v0 = 216 m/s - ϑ2 = 16° - T2 = calculated above

D2 = 216 * cos(16°) * T2 Since the projectiles collide with each other, the horizontal distances covered by both projectiles should be equal. Therefore, we can set up the following equation:

D1 = D2

Substituting the values of D1 and D2, we get:

216 * cos(55°) * T1 = 216 * cos(16°) * T2

Now, we can solve this equation to find the time interval between the two shots.

Calculation

Let's calculate the time interval between the two shots.

Using the equation T1 = (2 * 216 * sin(55°)) / g and T2 = (2 * 216 * sin(16°)) / g, we can substitute the values of T1 and T2 into the equation 216 * cos(55°) * T1 = 216 * cos(16°) * T2 and solve for the time interval.

Substituting the values, we get:

(216 * cos(55°) * (2 * 216 * sin(55°)) / g) = (216 * cos(16°) * (2 * 216 * sin(16°)) / g)

Simplifying the equation, we get:

cos(55°) * sin(55°) = cos(16°) * sin(16°)

Using the trigonometric identity sin(2ϑ) = 2 * sin(ϑ) * cos(ϑ), we can rewrite the equation as:

sin(110°) = sin(32°)

Since the sine function is an odd function, we can rewrite the equation as:

sin(110°) = sin(180° - 32°)

From trigonometric identities, we know that sin(180° - ϑ) = sin(ϑ). Therefore, the equation becomes:

sin(110°) = sin(32°)

Now, we can solve this equation to find the time interval between the two shots.

Answer

The time interval between the two shots, at which the projectiles collide with each other, can be found by solving the equation sin(110°) = sin(32°).

Please note that the solution to this equation involves finding the angles for which the sine function is equal. This can be done using a scientific calculator or mathematical software.