Question

Тело брошено горизонтально со скоростью 20 м/с. Определить перемещение тела от точки бросания, при котором скорость будет направлена под углом 45 градусов к горизонту

Answer 1

$r=\sqrt{x^2+y^2}.$

Движение по оси Х - равномерное:

$x=v_{0}*t.$

Движение по осиУ - равноускоренное с ускорением g и нулевой начальной скоростью:

$v_{y}=gt,\ \ v_{y}=v_{0}*tga=v_{0},\ \ \ t=\frac{v_{0}}{g}.$

$y=\frac{gt^2}{2}=\frac{v_{0}^2}{2g}.$

Тогда :

$x=\frac{v_{0}^2}{g}$

Находим модуль перемещения:

$r=\sqrt{\frac{v_{0}^4}{4g^2}+\frac{v_{0}^4}{g^2}}=\frac{v_{0}^2*\sqrt{5}}{2g}\approx45\ m.$

Ответ: 45 м.

Answer 2

Problem Analysis

We are given that an object is thrown horizontally with a velocity of 20 m/s. We need to determine the displacement of the object from the point of release when the velocity is directed at an angle of 45 degrees to the horizontal.

Solution

To solve this problem, we can use the equations of motion for projectile motion. Since the object is thrown horizontally, the initial vertical velocity is zero. The horizontal velocity remains constant throughout the motion.

Let's break down the problem into two components: horizontal and vertical.

# Horizontal Component

The horizontal component of velocity remains constant throughout the motion. Therefore, the displacement in the horizontal direction can be calculated using the equation:

Displacement in the horizontal direction = Horizontal velocity × Time

Since the object is thrown horizontally, the initial horizontal velocity is equal to the given velocity of 20 m/s.

# Vertical Component

The vertical component of velocity changes due to the effect of gravity. We can calculate the time of flight using the equation:

Time of flight = 2 × (Vertical velocity at 45 degrees) / (Acceleration due to gravity)

The vertical velocity at 45 degrees can be calculated using the equation:

Vertical velocity at 45 degrees = Initial velocity × sin(45 degrees)

The displacement in the vertical direction can be calculated using the equation:

Displacement in the vertical direction = (Vertical velocity at 45 degrees) × Time of flight

# Total Displacement

The total displacement can be calculated using the Pythagorean theorem:

Total displacement = √((Displacement in the horizontal direction)^2 + (Displacement in the vertical direction)^2)

Let's calculate the displacement using the given values.

Calculation

Given: Initial velocity (v0) = 20 m/s Angle (θ) = 45 degrees

Using the equations mentioned above, we can calculate the displacement.

Horizontal component: Horizontal velocity = Initial velocity = 20 m/s

Vertical component: Vertical velocity at 45 degrees = Initial velocity × sin(45 degrees) = 20 m/s × sin(45 degrees) = 20 m/s × 0.7071 ≈ 14.14 m/s

Time of flight = 2 × (Vertical velocity at 45 degrees) / (Acceleration due to gravity) = 2 × 14.14 m/s / 9.8 m/s² ≈ 2.88 s

Displacement in the vertical direction = (Vertical velocity at 45 degrees) × Time of flight = 14.14 m/s × 2.88 s ≈ 40.77 m

Total displacement = √((Displacement in the horizontal direction)^2 + (Displacement in the vertical direction)^2) = √((20 m/s × 2.88 s)^2 + (40.77 m)^2) ≈ 47.97 m

Answer

The displacement of the object from the point of release, when the velocity is directed at an angle of 45 degrees to the horizontal, is approximately 47.97 meters.

Please note that the above calculations assume no air resistance and a constant acceleration due to gravity.