Acceleration in 2D: Videos & Practice Problems

To calculate acceleration in two dimensions, use the equation $a^2=a^x+a^y$ and the components $a=\frac{\mathrm{\Delta v}}{\mathrm{\Delta t}}$. The angle is found using $\theta =\tan -1\frac{a}{x}$. Understanding these concepts is crucial for analyzing motion and changes in velocity effectively.

Acceleration in 2D

When calculating the acceleration of an object moving in two dimensions, it's essential to understand that acceleration can change both the magnitude and direction of an object's velocity. The fundamental equation for acceleration remains the same, expressed as:

\[ a = \frac{\Delta v}{\Delta t} \]

In two dimensions, acceleration can be broken down into its components along the x and y axes, denoted as \( a_x \) and \( a_y \). The relationship between the components and the overall acceleration can be established using the Pythagorean theorem:

\[ a = \sqrt{a_x^2 + a_y^2} \]

To find the angle of the acceleration vector, the tangent inverse function is utilized:

\[ \theta = \tan^{-1}\left(\frac{a_y}{a_x}\right) \]

For the components of acceleration, similar to velocity, we can express them in terms of the change in velocity over time:

\[ a_x = \frac{\Delta v_x}{\Delta t} \]

\[ a_y = \frac{\Delta v_y}{\Delta t} \]

Alternatively, if the magnitude of the acceleration and its direction are known, the components can be calculated using trigonometric functions:

\[ a_x = a \cdot \cos(\theta) \]

\[ a_y = a \cdot \sin(\theta) \]

To illustrate these concepts, consider a toy car that initially moves at 20 m/s along the x-axis and later moves at 67 m/s at an angle of 26.5 degrees. To find the components of acceleration, we first determine the final velocity components:

1. The initial velocity in the x-direction is \( v_{i_x} = 20 \, \text{m/s} \) and in the y-direction \( v_{i_y} = 0 \, \text{m/s} \).

2. The final velocity components can be calculated as:

\[ v_{f_x} = 67 \cdot \cos(26.5^\circ) \approx 60 \, \text{m/s} \]

\[ v_{f_y} = 67 \cdot \sin(26.5^\circ) \approx 30 \, \text{m/s} \]

Next, we can calculate the acceleration components:

\[ a_x = \frac{v_{f_x} - v_{i_x}}{\Delta t} = \frac{60 - 20}{10} = 4 \, \text{m/s}^2 \]

\[ a_y = \frac{v_{f_y} - v_{i_y}}{\Delta t} = \frac{30 - 0}{10} = 3 \, \text{m/s}^2 \]

With the components of acceleration determined, we can find the magnitude:

\[ a = \sqrt{4^2 + 3^2} = 5 \, \text{m/s}^2 \]

Finally, the direction of the acceleration can be calculated using the tangent inverse function:

\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 37^\circ \]

In summary, understanding the relationship between acceleration, velocity, and their components in two dimensions is crucial for analyzing motion effectively. By applying these principles, one can solve various problems involving acceleration in a systematic manner.

A football at rest is kicked by a football kicker. The ball is in contact with the kicker's foot for 0.050s, during which it experiences an acceleration a = 340 m/s². The ball is launched at an angle of 40° above the ground (x-axis). Calculate the horizontal and vertical components of the launch velocity.

• A. 13 m/s horizontal; 10.9 m/s vertical
• B. 130 m/s horizontal; 109 m/s vertical
• C. 10.9 m/s horizontal; 13 m/s vertical
• D. 17 m/s horizontal; 17 m/s vertical

Answer: (A)

Here’s what students ask on this topic:

How do you calculate acceleration in two dimensions?

To calculate acceleration in two dimensions, you can use the equation:

$a^2=a^x+a^y$

where $a$ is the acceleration vector, and $ax$ and $ay$ are its components in the x and y directions, respectively. Additionally, you can use the components of the change in velocity over time:

$a=\frac{\mathrm{\Delta v}}{\mathrm{\Delta t}}$

where $\mathrm{\Delta v}$ is the change in velocity and $\mathrm{\Delta t}$ is the change in time.

What is the formula for finding the angle of acceleration in two dimensions?

The angle of acceleration in two dimensions can be found using the inverse tangent function:

$\theta ={tan}^{-1}\frac{a}{y}$

where $\theta$ is the angle, $ay$ is the acceleration component in the y direction, and $ax$ is the acceleration component in the x direction.

How do you find the components of acceleration in the x and y directions?

To find the components of acceleration in the x and y directions, you can use the following equations:

$ax=\frac{\mathrm{\Delta v}}{x}$

$ay=\frac{\mathrm{\Delta v}}{y}$

where $\mathrm{\Delta v}x$ and $\mathrm{\Delta v}y$ are the changes in velocity in the x and y directions, respectively, and $\mathrm{\Delta t}$ is the change in time.

What is the relationship between acceleration and velocity in two dimensions?

In two dimensions, acceleration is the rate of change of velocity. It can change either the magnitude or the direction of the velocity, or both. The relationship is given by:

$a=\frac{\mathrm{\Delta v}}{\mathrm{\Delta t}}$

where $a$ is the acceleration, $\mathrm{\Delta v}$ is the change in velocity, and $\mathrm{\Delta t}$ is the change in time. The components of acceleration in the x and y directions can be found using:

$ax=\frac{\mathrm{\Delta v}}{x}$

$ay=\frac{\mathrm{\Delta v}}{y}$

How do you use the Pythagorean theorem to find the magnitude of acceleration in two dimensions?

To find the magnitude of acceleration in two dimensions using the Pythagorean theorem, you can use the following equation:

$a=\sqrt{a^x}$

where $a$ is the magnitude of the acceleration, and $ax$ and $ay$ are the components of acceleration in the x and y directions, respectively. This equation is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.