Question

Автомобиль движется по круговому арочному мосту радиусом 195 метров с постоянным касательным ускорением 2 м/с. В некоторой начальной точке он имел скорость 10 м/с. Определить полное ускорение автомобиля через 3 и 10 секунд после этого момента

Answer 1

Ответ:

а₃ ≈ 2,392 м/с²;    а₁₀ ≈ 5,030 м/с².

Объяснение:

R = 195 м

$a_{\tau} = 2$ м/с²

v₀ = 10 м/с

t₁ = 3 c

t₂ = 10 c

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a₃ и a₁₀ - ?

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В момент времени t₁ = 3 c скорость автомобиля

$v_3 = v_0 + a_{\tau}\cdot t_1 = 10 + 2 \cdot 3 = 16$ (м/с)

Нормальное ускорение автомобиля

$a_{3n} = \dfrac{v_3^2}{R} } = \dfrac{16^2}{195} } \approx 1.313 ~(m/s^2)$

Полное ускорение автомобиля

$a_3 = \sqrt{a_{3n}^2 + a_{\tau}^2} = \sqrt{1.313^2 + 2^2} \approx 2.392~(m/s^2) .$

В момент времени t₂ = 10 c скорость автомобиля

$v_{10} = v_0 + a_{\tau}\cdot t_2 = 10 + 2 \cdot 10 = 30$ (м/с)

Нормальное ускорение автомобиля

$a_{10n} = \dfrac{v_{10}^2}{R} } = \dfrac{30^2}{195} } \approx 4.615 ~(m/s^2)$

Полное ускорение автомобиля

$a_{10} = \sqrt{a_{10n}^2 + a_{\tau}^2} = \sqrt{4.615^2 + 2^2} \approx 5.030~(m/s^2) .$

Answer 2

Calculating the Acceleration of a Car on a Circular Bridge

To calculate the total acceleration of the car at 3 and 10 seconds after the initial point, we can use the following steps:

1. Calculate the Tangential Acceleration: - The tangential acceleration of the car can be calculated using the formula: \[ a_t = \frac{dv}{dt} \] where \( v \) is the velocity of the car.

2. Calculate the Radial Acceleration: - The radial acceleration of the car can be calculated using the formula: \[ a_r = \frac{v^2}{r} \] where \( v \) is the velocity of the car and \( r \) is the radius of the circular bridge.

3. Calculate the Total Acceleration: - The total acceleration of the car can be calculated using the formula: \[ a = \sqrt{a_t^2 + a_r^2} \]

Calculation at 3 Seconds:

1. Tangential Acceleration at 3 Seconds: - The tangential acceleration at 3 seconds can be calculated using the initial velocity and the given tangential acceleration.

2. Radial Acceleration at 3 Seconds: - The radial acceleration at 3 seconds can be calculated using the initial velocity and the radius of the circular bridge.

3. Total Acceleration at 3 Seconds: - The total acceleration at 3 seconds can be calculated using the tangential and radial accelerations at 3 seconds.

Calculation at 10 Seconds:

1. Tangential Acceleration at 10 Seconds: - The tangential acceleration at 10 seconds can be calculated using the initial velocity and the given tangential acceleration.

2. Radial Acceleration at 10 Seconds: - The radial acceleration at 10 seconds can be calculated using the initial velocity and the radius of the circular bridge.

3. Total Acceleration at 10 Seconds: - The total acceleration at 10 seconds can be calculated using the tangential and radial accelerations at 10 seconds.

Let's calculate the total acceleration of the car at 3 and 10 seconds after the initial point using the provided information.