The motion of a particle of mass $m$ is described by $y = ut + \frac{1}{2}gt^{2}$. Find the force acting on the particle.

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> [!TLDR]
> To find the force, we first determine the acceleration by differentiating the given position equation twice with respect to time. Then, we apply Newton's Second Law, $F=ma$. The force acting on the particle is $\boxed{mg}$.

The problem asks us to find the force acting on a particle whose motion is described by the equation $y = ut + \frac{1}{2}gt^{2}$. The mass of the particle is given as $m$.

The fundamental principle connecting force and motion is **Newton's Second Law**, which states that the net force acting on an object is equal to the product of its mass and acceleration ($F=ma$). To find the force, we need to determine the acceleration of the particle.

We are given the position of the particle, $y$, as a function of time, $t$. We know that:
1.  **Velocity** is the rate of change of position with respect to time. Mathematically, $v = \frac{dy}{dt}$.
2.  **Acceleration** is the rate of change of velocity with respect to time. Mathematically, $a = \frac{dv}{dt}$, or equivalently, $a = \frac{d^{2}y}{dt^{2}}$.

Therefore, our strategy will be to differentiate the given position equation twice to find the acceleration, and then use Newton's Second Law to calculate the force.

1.  **Identify the given position function:**
    The motion of the particle is described by the equation:
    $$y(t) = ut + \frac{1}{2}gt^{2}$$
    Here, $u$ represents the initial velocity and $g$ represents the acceleration due to gravity. Both $u$ and $g$ are constants in this context.

2.  **Calculate the velocity of the particle:**
    Velocity $v(t)$ is the first derivative of the position function $y(t)$ with respect to time $t$.
    $$v(t) = \frac{dy}{dt}$$
    Differentiating $y(t) = ut + \frac{1}{2}gt^{2}$:
    $$\frac{d}{dt}(ut) = u \cdot \frac{d}{dt}(t) = u \cdot 1 = u$$
    $$\frac{d}{dt}\left(\frac{1}{2}gt^{2}\right) = \frac{1}{2}g \cdot \frac{d}{dt}(t^{2}) = \frac{1}{2}g \cdot (2t) = gt$$
    Combining these, the velocity function is:
    $$v(t) = u + gt$$

3.  *

Q.The motion of a particle of mass $m$ is described by $y = ut + \frac{1}{2}gt^{2}$ . Find the force acting on the particle.

Concept understanding — Newton Second Law

Newton's Second Law: The Law That Connects Force and Motion

The Intuition First

The Precise Statement

What This Really Means

A Simple Example

Concept: Differentiating Position to get Acceleration, then Newton's Second Law

Q.The motion of a particle of mass mmm is described by y=ut+12gt2y = ut + \frac{1}{2}gt^{2}y=ut+21​gt2. Find the force acting on the particle.

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Q.The motion of a particle of mass $m$ is described by $y = ut + \frac{1}{2}gt^{2}$ . Find the force acting on the particle.

Unlock everything free for 14 days