Newtons Laws Of Motion

We are given the situation of a body with a temperature slightly higher than the surrounding temperature . The rate of cooling, , is plotted against the temperature of the body . % Step 1: Understanding the Law of Cooling According to Newton's law of cooling, the rate of change of temperature with respect to time is proportional to the difference between the body's temperature and the surrounding temperature . This can be written as: where: - is the constant of proportionality, - is the surrounding temperature, - is the temperature of the body. % Step 2: Interpreting the Cooling Rate As the body cools, the difference becomes smaller. Initially, when is much higher than , the rate of cooling is large. As the temperature of the body approaches , the rate of cooling decreases, since the difference between the body’s temperature and the surrounding temperature is smaller. This behavior leads to a curve that initially has a steep negative slope but gradually flattens out as the temperature difference reduces. This implies the rate of cooling decreases as the body’s temperature approaches that of the surroundings. % Step 3: Conclusion The relationship between and would, therefore, follow a curve that decreases with a negative slope, which corresponds to option (C).

Step 1: Understanding the Force Equation The force acting on the body is given as: Where: - are constants representing the components of the force in the and directions, - are unit vectors along the and axes, respectively. Step 2: Newton's Second Law of Motion From Newton's second law, we know that: Where: - is the acceleration vector, and - is the mass of the body. The acceleration is the derivative of velocity with respect to time: And velocity is the derivative of displacement: Step 3: Integrating to Find Displacement To find the displacement , we need to integrate the acceleration twice because acceleration is the second derivative of displacement with respect to time: Since , the acceleration is: The displacement is the integral of acceleration over time. The solution after two integrations (since the body starts at rest at the origin) will give the coordinates as: Step 4: Conclusion Therefore, the coordinates of the body after time are given by: