Motion in Two Dimensions-Motion in a Plane - Acceleration in a Plane

Acceleration describes how quickly the velocity of a moving object changes. Just like velocity can be measured over a time interval or at a specific instant, acceleration also has two types: average acceleration and instantaneous acceleration. In two-dimensional motion, both types are treated the same way as in one-dimensional motion, but we must consider components in both the x and y directions. Understanding acceleration helps us describe how objects speed up, slow down, or change direction during their motion.x

Average Acceleration (\[\vec a_{av}\]): The change in velocity divided by the time interval over which that change occurs. It represents the acceleration over a specific time period between two moments.

Instantaneous Acceleration (\[\vec a\]): The acceleration of an object at a specific instant of time. It is the limit of average acceleration as the time interval approaches zero, or the rate of change of velocity with respect to time.

The average acceleration between t₁​ and t₂ is

\[\vec{a}_{av}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1}\]

That means: “Average acceleration = Change in velocity ÷ Time interval”

In component form: 
\[\vec{a}_{av}=\left(\frac{v_{2x}-v_{1x}}{t_2-t_1}\right)\hat{i}+\left(\frac{v_{2y}-v_{1y}}{t_2-t_1}\right)\hat{j}\]

Magnitude:
\[a_{av}=\sqrt{(a_{av,x})^2+(a_{av,y})^2}\]

Direction:
\[\tan\theta=\frac{a_{av,y}}{a_{av,x}}\]

When the time interval becomes extremely small (approaches zero), the average acceleration becomes the instantaneous acceleration — i.e., acceleration at a specific instant.

\[\vec{a}=\lim_{\Delta t\to0}\frac{\Delta\vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]

Or in components:
\[\vec{a}=\left(\frac{dv_x}{dt}\right)\hat{i}+\left(\frac{dv_y}{dt}\right)\hat{j}\]

Since \[v_x=\frac{dx}{dt}\mathrm{~and~}v_y=\frac{dy}{dt}\mathrm{:}\]
\[\vec{a}=\left(\frac{d^2x}{dt^2}\right)\hat{i}+\left(\frac{d^2y}{dt^2}\right)\hat{j}\]

Magnitude:
a = \[\sqrt{\left(\frac{d^2x}{dt^2}\right)^2+\left(\frac{d^2y}{dt^2}\right)^2}\]

Direction:
\[\tan\theta=\frac{dv_y/dt}{dv_x/dt}=\frac{dv_y}{dv_x}\]

• Vector Quantity: Acceleration has both magnitude and direction.
• Component Form: Can be expressed separately in the x and y directions.
• Derivative Relationship: Instantaneous acceleration is the derivative of velocity with respect to time.
• Second Derivative: Acceleration is the second derivative of position with respect to time.
• Independent Components: The x and y components of acceleration are independent of each other.
• Slope Representation: The direction of instantaneous acceleration is the slope of the tangent to the velocity graph (plot of v_y versus v_x).

• Understanding Motion: Helps describe how an object's motion changes in two dimensions.
• Predicting Future Motion: Knowing acceleration allows us to predict where an object will be at future times.
• Force Analysis: Newton's second law connects force to acceleration, making it essential for understanding dynamics.
• Practical Applications: Used in vehicle motion, projectile motion, circular motion, and many real-world scenarios.
• Mathematical Tool: Differentiating position gives velocity; differentiating velocity gives acceleration—a fundamental calculus application in physics.
• Component Analysis: Breaking acceleration into x and y components simplifies problem-solving in 2D motion.

Problem: The position vectors of three particles are given by:

• \[\vec x_1\] = (5\[\hat i\] + 5\[\hat j\]) m
• \[\vec x_2\] = (5t\[\hat i\] + 5t\[\hat j\]) m
• \[\vec x_3\] = (5t\[\hat i\] + 10t²\[\hat j\]) m

Determine the velocity and acceleration for each in SI units.

Solution:

Particle 1:

\[\vec v_1\] =  = 0 (since \[\vec x_1\] does not depend on time)

• The particle is at rest (stationary).

Particle 2:

\[\vec{v}_2=\frac{d\vec{x}_2}{dt}=5\hat{i}+5\hat{j}\mathrm{~m/s}\]

• Velocity is constant and does not change with time, so acceleration is zero: \[\vec a_2\] = 0
• Magnitude: v₂ = \[\sqrt{5^2+5^2}\]= \[\sqrt {50}\] $\approx 7.07$ m/s
• Direction: tan⁡ θ = \[\frac {5}{}\] = 1, so θ = 45° to the horizontal
• Particle 2 moves at constant velocity in a straight line at 45° to the horizontal.

Particle 3:

\[\vec{v}_3=\frac{d\vec{x}_3}{dt}=5\hat{i}+20t\hat{j}\mathrm{~m/s}\]

• Magnitude: v₃ = \[\sqrt{5^2+(20t)^2}=\sqrt{25+400t^2}\mathrm{~m/s}\]
• Direction: θ = tan⁡⁻¹ (\[\frac {20t}{5}\]) = tan^⁡−1 (4t) with the horizontal

\[\vec a_3\] = \[\frac{d\vec{v}_3}{dt}\] = 20\[\hat j\] m/s²

• The particle accelerates only in the y-direction at a constant rate of 20 m/s².
• The x-component of velocity remains constant at 5 m/s.
• The y-component increases linearly with time.

1. Driving a Car: When you press the accelerator, the car's velocity increases. The acceleration has components in different directions depending on whether you're speeding up on a straight road or turning a curve.
2. Throwing a Ball: When you throw a ball at an angle, it has acceleration due to gravity acting downward (y-component) while moving horizontally (x-component). The instantaneous acceleration at any moment is the vector sum of these components.
3. Airplane Taking Off: An airplane accelerates along the runway (x-direction) while climbing into the air (y-direction). The overall acceleration is the combination of these two components.
4. Roller Coaster: At different points on a roller coaster, the acceleration changes direction and magnitude. At the top of a loop, acceleration points downward; on a straight section, it might point forward.
5. Satellite Orbiting Earth: A satellite in orbit has instantaneous acceleration toward Earth's center. Even if its speed is constant, the direction constantly changes, resulting in continuous acceleration.