Step 1: Identifying Variables and Initial Conditions

Given data:

• Mass of the motorcar ($m$) = $1200\text{ kg}$
• Acceleration with three persons ($a$) = $0.2\text{ m s}^{-2}$
• Let the force exerted by one person be $F$.
• Let the frictional force opposing the motion be $f$.

[Per Newton's First Law of Motion, an object moving at a constant velocity implies that the net external force acting on the object is zero. Therefore, the force applied must be equal to the frictional force.]

Step 2: Analyzing the Two-Person Scenario

When two persons push the car at a uniform velocity, the total applied force equals the frictional force:

$2F - f = 0 \implies f = 2F$ [Equation 1]

Step 3: Analyzing the Three-Person Scenario

When three persons push the car, it produces an acceleration ($a$). [Per Newton's Second Law of Motion: $F_{net} = ma$]:

$3F - f = ma$ [Equation 2]

Step 4: Solving the System of Equations

Substitute Equation 1 into Equation 2:

$3F - (2F) = ma$

$F = ma$

Substitute the given values ($m = 1200\text{ kg}$, $a = 0.2\text{ m s}^{-2}$):

$F = 1200\text{ kg} \times 0.2\text{ m s}^{-2}$

$F = 240\text{ N}$

Visual Representation of Force Vectors

Step 5: Verification of Results

Final Solution: Each person pushes the motorcar with a force of 240 N.

Step 1: Analyzing the Provided Data

Given the distance ($s$) and time ($t$) values provided in the table:

Step 2: Mathematical Evaluation of the Motion

To determine the nature of the acceleration, we evaluate the relationship between distance ($s$) and time ($t$). Observing the numerical pattern:

• At $t=1, s=1^3=1$
• At $t=2, s=2^3=8$
• At $t=3, s=3^3=27$
• At $t=4, s=4^3=64$

The motion follows the cubic function: $s(t) = kt^3$, where $k=1 \text{ m/s}^3$. [Based on polynomial regression analysis of the data points].

Step 3: Deriving Velocity and Acceleration

To find velocity ($v$), we calculate the first derivative of distance with respect to time [Per the definition of instantaneous velocity, $v = \frac{ds}{dt}$]:

$v = \frac{d}{dt}(t^3) = 3t^2$

To find acceleration ($a$), we calculate the derivative of velocity with respect to time [Per the definition of instantaneous acceleration, $a = \frac{dv}{dt}$]:

$a = \frac{d}{dt}(3t^2) = 6t$

Step 4: Visualizing the Motion Profile

Step 5: Conclusion on Acceleration

Since the derived expression for acceleration is $a = 6t$, the value of acceleration is directly proportional to time ($t$). As $t$ increases, $a$ increases linearly. Therefore, the acceleration is not constant, nor is it zero; it is increasing.

Final Solution: The acceleration of the object is increasing, as the velocity is changing at a non-uniform rate indicated by the cubic relationship between distance and time.

Given Variables & Initial Setup

To determine the force exerted by the nail on the hammer, we utilize Newton's Second Law of Motion. First, we identify the physical quantities provided in the SI system:

• Mass of the hammer ($m$) = $500 \text{ g} = 0.5 \text{ kg}$ [Conversion factor: $1 \text{ kg} = 1000 \text{ g}$]
• Initial velocity of the hammer ($u$) = $50 \text{ m s}^{-1}$
• Final velocity of the hammer ($v$) = $0 \text{ m s}^{-1}$ [Since the nail stops the hammer]
• Time taken to stop ($t$) = $0.01 \text{ s}$

Step 1: Calculating the Change in Momentum

Momentum ($p$) is defined as the product of mass and velocity. We calculate the initial momentum ($p_i$) and final momentum ($p_f$):

$p_i = m \times u = 0.5 \text{ kg} \times 50 \text{ m s}^{-1} = 25 \text{ kg m s}^{-1}$

$p_f = m \times v = 0.5 \text{ kg} \times 0 \text{ m s}^{-1} = 0 \text{ kg m s}^{-1}$

Change in momentum ($\Delta p$) = $p_f - p_i = 0 - 25 = -25 \text{ kg m s}^{-1}$

Step 2: Applying Newton's Second Law

Per Newton’s Second Law, the force ($F$) exerted is equal to the rate of change of momentum:

$F = \frac{\Delta p}{t}$ [By definition of Force, $F = ma = m \frac{(v - u)}{t}$]

$F = \frac{-25 \text{ kg m s}^{-1}}{0.01 \text{ s}}$

$F = -2500 \text{ N}$

Step 3: Interpretation of Results

The negative sign indicates that the direction of the force is opposite to the initial direction of the hammer's motion. The magnitude of the force exerted by the nail on the hammer is $2500 \text{ N}$.

Final Solution: The force exerted by the nail on the hammer is 2500 N in the direction opposite to the motion of the hammer.

Step 1: Given Variables & Unit Conversions

First, we identify the physical quantities provided and convert them into SI units (kilograms for mass, meters per second for velocity, and seconds for time).

• Mass ($m$) = $1200\text{ kg}$
• Initial velocity ($u$) = $90\text{ km/h} = 90 \times \frac{5}{18}\text{ m/s} = 25\text{ m/s}$
• Final velocity ($v$) = $18\text{ km/h} = 18 \times \frac{5}{18}\text{ m/s} = 5\text{ m/s}$
• Time interval ($t$) = $4\text{ s}$

Step 2: Calculating Acceleration

Acceleration ($a$) is defined as the rate of change of velocity [Per Newton’s First Law of Motion and Kinematic Definitions].

$a = \frac{v - u}{t}$

$a = \frac{5\text{ m/s} - 25\text{ m/s}}{4\text{ s}}$

$a = \frac{-20\text{ m/s}}{4\text{ s}} = -5\text{ m/s}^2$

[The negative sign indicates retardation or deceleration].

Step 3: Calculating Change in Momentum

Momentum ($p$) is the product of mass and velocity. The change in momentum ($\Delta p$) is given by $m(v - u)$ [Per Newton’s Second Law of Motion].

$\Delta p = m \times v - m \times u$

$\Delta p = 1200\text{ kg} \times (5\text{ m/s} - 25\text{ m/s})$

$\Delta p = 1200\text{ kg} \times (-20\text{ m/s})$

$\Delta p = -24000\text{ kg}\cdot\text{m/s}$

Step 4: Calculating Magnitude of the Force

The magnitude of the force ($F$) required is calculated using the formula $F = ma$ [Per Newton’s Second Law of Motion: $F = \frac{\Delta p}{\Delta t}$].

$F = 1200\text{ kg} \times (-5\text{ m/s}^2)$

$F = -6000\text{ N}$

The magnitude of the force is $6000\text{ N}$ (acting in the direction opposite to the motion).

Visual Representation of Velocity Change

Final Solution: The acceleration is $-5\text{ m/s}^2$, the change in momentum is $-24000\text{ kg}\cdot\text{m/s}$, and the magnitude of the force required is $6000\text{ N}$.

Step 1: Analyzing the Kinematic Data

To infer the forces acting on the object, we must first analyze the nature of the motion by calculating the speed ($v$) at different time intervals. The object's motion is defined by the provided table (Time $t$ in seconds, Distance $s$ in meters):

Step 2: Mathematical Derivation of Motion Type

We observe that the distance traveled $s$ is proportional to the cube of time $t$. Specifically, $s = k \cdot t^3$. Let us verify the proportionality constant ($k$):

• At $t = 2, s = 1 \implies 1 = k(2^3) \implies k = 1/8$
• At $t = 4, s = 8 \implies 8 = k(4^3) \implies 8 = k(64) \implies k = 1/8$
• At $t = 10, s = 125 \implies 125 = k(10^3) \implies 125 = k(1000) \implies k = 1/8$

Since $s = \frac{1}{8} t^3$, the velocity ($v$) is the first derivative of displacement with respect to time [Per the definition of instantaneous velocity]:

$v = \frac{ds}{dt} = \frac{d}{dt}(\frac{1}{8} t^3) = \frac{3}{8} t^2$

The acceleration ($a$) is the first derivative of velocity with respect to time [Per the definition of instantaneous acceleration]:

$a = \frac{dv}{dt} = \frac{d}{dt}(\frac{3}{8} t^2) = \frac{3}{4} t$

Step 3: Inference Regarding Forces

Since the acceleration $a = 0.75t$ is not constant and increases linearly with time, we apply Newton's Second Law of Motion ($F = ma$):

Given that the mass ($m$) of the object is constant, if the acceleration is changing, the net external force ($F$) acting on the object must also be changing linearly with time [Per Newton’s Second Law of Motion: $F_{net} = m \cdot a(t)$].

Step 4: Conclusion

Because the acceleration is non-zero and non-uniform, the object is undergoing non-uniform acceleration. This implies that there is an unbalanced, time-dependent force acting upon the object. A state of constant velocity (which would imply zero net force) is not maintained, and the acceleration is not constant (which would imply a constant force). Therefore, the force must be varying to satisfy the condition of $F(t) = m \cdot (\frac{3}{4}t)$.

Final Solution: The object is moving with non-uniform acceleration; therefore, there is an unbalanced, time-varying external force acting on the object.