A particle has an initial velocity 3i + 4j and an acceleration of 0.4i + 0.3j. Its speed after 10 s

Q: Why do we add the $\hat{i}$ and $\hat{j}$ components separately?

In two-dimensional motion, the horizontal ($x$-axis, denoted by $\hat{i}$) and vertical ($y$-axis, denoted by $\hat{j}$) motions are completely independent of each other. Therefore, we must add the $x$-components together and the $y$-components together.

Q: What is the difference between velocity and speed in this context?

Velocity is a vector quantity, meaning it has both magnitude and direction (e.g., $7\hat{i} + 7\hat{j}$). Speed is a scalar quantity, which is simply the magnitude of the velocity vector (e.g., $7\sqrt{2}$). The question specifically asked for speed.

Q: Can we find the magnitude of $\mathbf{u}$ and $\mathbf{a}$ first, and then use the scalar equation $v = u + at$?

Absolutely not! You can only use the scalar equation $v = u + at$ if the initial velocity and acceleration are in the exact same direction. Since $\mathbf{u}$ and $\mathbf{a}$ point in different directions here, you must perform the vector addition first before finding the magnitude.

Q: How did $0.4\hat{i} + 0.3\hat{j}$ become $4\hat{i} + 3\hat{j}$?

According to the equation $\mathbf{v} = \mathbf{u} + \mathbf{a}t$, we must multiply the acceleration vector $\mathbf{a}$ by the time $t = 10\text{ s}$. Distributing the $10$ gives $0.4 \times 10 = 4$ and $0.3 \times 10 = 3$.

Q: Is the acceleration constant in this problem?

Yes. The given acceleration vector is $0.4\hat{i} + 0.3\hat{j}$. Because it consists only of constant numbers and does not contain any time variable $t$, the acceleration remains constant throughout the entire motion.

Question

A particle has an initial velocity $3\hat{i} + 4\hat{j}$ and an acceleration of $0.4\hat{i} + 0.3\hat{j}$. Its speed after $10	ext{ s}$ is

Options:

(A) 10 units

(B) $7\sqrt{2}$ units

(C) 7 units

(D) 8.5 units

Sigma Percentile · Accepted Answer

### Visualizing the Particle's Journey

Imagine a particle moving in a two-dimensional plane. We are given its initial velocity vector, $\mathbf{u} = 3\hat{i} + 4\hat{j}$. This tells us that initially, the particle is moving $3$ units along the x-axis and $4$ units along the y-axis every second.

Simultaneously, a constant acceleration is acting on it, given by $\mathbf{a} = 0.4\hat{i} + 0.3\hat{j}$. This acceleration continuously modifies the particle's velocity over time. Our objective is to determine the particle's **speed** exactly $10	ext{ s}$ after it starts moving.

### The Master Equation of Kinematics

To find the final velocity, we need a relationship that connects initial velocity, acceleration, and time. Since the acceleration is constant, we can confidently apply the first equation of motion in its vector form:

$$\mathbf{v} = \mathbf{u} + \mathbf{a}t$$

This elegant equation states that the final velocity is simply the initial velocity plus the total change in velocity caused by the acceleration over the given time interval.

### Executing the Vector Math

Let's carefully substitute the given values into our equation. We plug in the initial velocity vector, the acceleration vector, and the time $t = 10	ext{ s}$:

$$\mathbf{v} = (3\hat{i} + 4\hat{j}) + (0.4\hat{i} + 0.3\hat{j}) 	imes 10$$

Notice how the acceleration scales up over these ten seconds. Multiplying the acceleration vector by $10$ shifts the decimal point, giving us the total change in velocity:

$$\mathbf{v} = 3\hat{i} + 4\hat{j} + 4\hat{i} + 3\hat{j}$$

Now, we combine the corresponding components. In 2D kinematics, the horizontal ($x$) and vertical ($y$) motions are completely independent. Adding the $\hat{i}$ components ($3 + 4$) gives $7\hat{i}$, and adding the $\hat{j}$ components ($4 + 3$) gives $7\hat{j}$.

$$\mathbf{v} = 7\hat{i} + 7\hat{j}$$

### Finding the Speed

We have successfully found the final velocity vector, but there is a catch! The question specifically asks for the **speed**, not the velocity vector. Speed is a scalar quantity representing the magnitude of the velocity vector.

To find the magnitude, we use the Pythagorean theorem on the components of the velocity vector:

$$	ext{Speed} = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$$

Substituting our components:

$$|\mathbf{v}| = \sqrt{7^2 + 7^2}$$

$$|\mathbf{v}| = \sqrt{49 + 49} = \sqrt{98}$$

This simplifies beautifully to:

$$|\mathbf{v}| = 7\sqrt{2}	ext{ units}$$

And there we have it! The speed of the particle after $10	ext{ s}$ is $7\sqrt{2}	ext{ units}$.

Animated Solution for Physics - Kinematics: A particle has an initial velocity $3 \hat{i} + 4 \hat{j}$ and an acceleration of $0.4 \hat{i} + 0.3 \hat{j}$ . Its speed after $10 s$ is

Select Answer:

Visualized Solution

\text{Initial Setup}

\text{Equation of Motion}

\text{Substituting Values}

\text{Calculating Final Velocity}

\text{Magnitude of Velocity (Speed)}

\text{Further Analysis}

The Sigma Insight: Motion in a Plane

Visualizing the Particle's Journey

The Master Equation of Kinematics

Executing the Vector Math

Finding the Speed

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Animated Solution for Physics - Kinematics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle has an initial velocity 3i^+4j^​ and an acceleration of 0.4i^+0.3j^​. Its speed after 10 s is