Question

Камень брошен под углом 45° к горизонту со скоростью 10м/с.Через какое время вектор его скорости будет направлен под углом 30° к горизонту?

Answer 1

Найдем проекции скорости на оси x и y: Vxo = Vo * cos45, Vyo = Vo * sin 45. По оси х ускорение не действует, поэтому Vx = Vxo = Vo * cos45, а вот по оси у действует ускорение свободного падения g, поэтому Vy = Vyo - gt = Vo * sin45 - gt. Tg30 = (Vo sin45 - gt)/Vo cos 45; t = (Vosin45 - Vocos45 * tg 30)/g = 0,3 с. Когда gt становиться больше, чем Vosin45, камень начинает лететь вниз, тогда Tg30 = (gt - Vosin45)/Vocos45; t =( Vosin45 + Vocos45 * tg30)/g = 1,1 c. Итак первый раз скорость направлена под кглом 30 градусов во время 0,3 с, а во второй 1,1 с.

Answer 2

Problem Analysis

We are given that a stone is thrown at an angle of 45° to the horizon with a speed of 10 m/s. We need to find the time at which the velocity vector of the stone will be directed at an angle of 30° to the horizon.

Solution

To solve this problem, we can use the equations of motion for projectile motion. The horizontal and vertical components of the velocity can be calculated using trigonometry.

Let's break down the problem into steps:

1. Find the horizontal and vertical components of the initial velocity. 2. Determine the time at which the vertical component of the velocity becomes equal to the horizontal component. 3. Calculate the total time it takes for the velocity vector to be directed at an angle of 30° to the horizon.

Step 1: Find the horizontal and vertical components of the initial velocity

The initial velocity of the stone can be broken down into horizontal and vertical components using trigonometry. The horizontal component (Vx) can be calculated using the formula Vx = V * cos(θ), and the vertical component (Vy) can be calculated using the formula Vy = V * sin(θ), where V is the magnitude of the initial velocity and θ is the angle of projection.

Given: - Initial speed (V) = 10 m/s - Angle of projection (θ) = 45°

Using the above formulas, we can calculate the horizontal and vertical components of the initial velocity:

Vx = V * cos(θ) = 10 * cos(45°) = 10 * 0.7071 ≈ 7.071 m/s. Vy = V * sin(θ) = 10 * sin(45°) = 10 * 0.7071 ≈ 7.071 m/s.

So, the horizontal component of the initial velocity (Vx) is approximately 7.071 m/s, and the vertical component (Vy) is approximately 7.071 m/s.

Step 2: Determine the time at which the vertical component of the velocity becomes equal to the horizontal component

To find the time at which the vertical component of the velocity becomes equal to the horizontal component, we can use the equation of motion for vertical displacement:

Vy = Vy0 + gt,

where Vy is the vertical component of the velocity, Vy0 is the initial vertical component of the velocity, g is the acceleration due to gravity (approximately 9.8 m/s²), and t is the time.

Since the stone is thrown horizontally, the initial vertical component of the velocity (Vy0) is 0 m/s.

Using the equation Vy = Vy0 + gt, we can solve for t:

0 + 9.8t = 7.071.

Simplifying the equation, we get:

9.8t = 7.071.

Solving for t, we find:

t = 7.071 / 9.8 ≈ 0.721 seconds.

So, the time at which the vertical component of the velocity becomes equal to the horizontal component is approximately 0.721 seconds.

Step 3: Calculate the total time it takes for the velocity vector to be directed at an angle of 30° to the horizon

To find the total time it takes for the velocity vector to be directed at an angle of 30° to the horizon, we can use the equation of motion for vertical displacement:

y = y0 + Vyt - (1/2)gt²,

where y is the vertical displacement, y0 is the initial vertical displacement, Vy is the vertical component of the velocity, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s²).

Since the stone is thrown horizontally, the initial vertical displacement (y0) is 0 m.

Using the equation y = y0 + Vyt - (1/2)gt², we can solve for t:

0 + 7.071t - (1/2) * 9.8 * t² = 0.

Simplifying the equation, we get:

7.071t - 4.9t² = 0.

Factoring out t, we get:

t(7.071 - 4.9t) = 0.

Solving for t, we find two solutions:

t = 0 (which corresponds to the initial time when the stone is thrown) or t = 7.071 / 4.9 ≈ 1.445 seconds.

So, the total time it takes for the velocity vector to be directed at an angle of 30° to the horizon is approximately 1.445 seconds.

Answer

The time at which the vector of the stone's velocity will be directed at an angle of 30° to the horizon is approximately 1.445 seconds.

Please note that the above calculations are based on the given information and the assumptions made.