Question

Камень брошен со скоростью 20м/с под углом 30 градусов к горизонту с башни высотой 15 м. На каком расстоянии от основания башни камень упадет на поверхность Земли?

Answer 1

$y = y_{0} + v_{0_{y}}t + \frac{at^2}{2}$
Учитывая, что по вертикальной оси ускорение $a = -g$, высота $y_{0} = 15$ и то, что мы ищем момент столкновения с землёй, получаем формулу 
$0 = 15 + v_{0_{y}}t - \frac{gt^2}{2}$
$0 = 15 + v_{0}*sin\alpha*t - \frac{gt^2}{2}$ (т. к. $v_{0_{y}}=v_{0}*sin\alpha)$
$0 = 15 + 20*sin(30)*t - \frac{10t^2}{2} \\ 10t^2-20t-30=0 \\ t^2-2t-3=0$
Решая квадратное уравнение получим, что t = 3с
$x = x_{0} + v_{0_{x}}t + \frac{at^2}{2} \\ x = v_{0_{x}}t \\ x = v_{0}*cos\alpha*t$
$x = 20*cos(30)*3 \\ x = 10 \sqrt{3} *3 \\ x =30\sqrt{3}$
$x$ ≈ 52м
Ответ: 52 метра.

Answer 2

Calculation of the Distance

To calculate the distance at which the stone will hit the surface of the Earth, we can use the equations of projectile motion. The horizontal distance traveled by the stone can be calculated using the formula:

Range = (Initial Velocity * Time of Flight * Cosine of Launch Angle)

The time of flight can be calculated using the formula:

Time of Flight = (2 * Initial Velocity * Sine of Launch Angle) / Acceleration due to Gravity

In this case, the initial velocity of the stone is 20 m/s and the launch angle is 30 degrees. The acceleration due to gravity is approximately 9.8 m/s^2.

Let's calculate the distance:

1. Calculate the time of flight: - Initial Velocity = 20 m/s - Launch Angle = 30 degrees - Acceleration due to Gravity = 9.8 m/s^2

Time of Flight = (2 * 20 * Sin(30)) / 9.8

2. Calculate the range: - Initial Velocity = 20 m/s - Time of Flight = Calculated in step 1 - Launch Angle = 30 degrees

Range = (20 * Time of Flight * Cos(30))

Let's calculate the values using the given formulas:

1. Calculate the time of flight: - Initial Velocity = 20 m/s - Launch Angle = 30 degrees - Acceleration due to Gravity = 9.8 m/s^2

Time of Flight = (2 * 20 * Sin(30)) / 9.8

Plugging in the values: Time of Flight = (2 * 20 * 0.5) / 9.8 = 2.04 seconds

2. Calculate the range: - Initial Velocity = 20 m/s - Time of Flight = 2.04 seconds - Launch Angle = 30 degrees

Range = (20 * 2.04 * Cos(30))

Plugging in the values: Range = (20 * 2.04 * 0.866) = 35.36 meters

Therefore, the stone will hit the surface of the Earth at a distance of approximately 35.36 meters from the base of the tower.