Chapter 7, Problem 1FC

To determine

To choose: The correct option from the given options.

Answer to Problem 1FC

The correct option is, “(b). They have the same kinetic energies, but different momenta”.

Explanation of Solution

Given data:

Two identical cars travel at the same speed. One car travels towards the east and the other car travel towards the north.

Formula used:

Write the expression of kinetic energy $\left(\text{K}\right)$ of the object as follows:

$K=\frac{1}{2}m{\left(\overrightarrow{\text{v}}\right)}^2$ (1)

Here,

$m$ is mass of the object, and

$\overrightarrow{\text{v}}$ is velocity of the object .

From the formula in Equation (1), the kinetic energy depends on the mass and velocity of the object.

As both the cars are identical in nature, the mass of both the cars is the same. From the given data, both the cars travel at the same speed.

As both the parameters (mass and velocity) are the same for both the cars, from the expression in Equation (1), the kinetic energy is the same for both the cars.

Write the expression for linear momentum $\left(\overrightarrow{\text{p}}\right)$ of the object as follows:

$\overrightarrow{\text{p}}=m\overrightarrow{\text{v}}$ (2)

From the formula in Equation (2), the momentum of the object depends on the mass and velocity of the object.

The velocity $\left(\overrightarrow{\text{v}}\right)$, and momentum $\left(\overrightarrow{\text{p}}\right)$ are vector quantities (having both magnitude and direction). However, the kinetic energy is a scalar quantity (having only magnitude) since the ${\left(\overrightarrow{\text{v}}\right)}^2$ is a scalar quantity.

As the momentum is a vector quantity, the momentum depends on the direction of the cars also. From the given data, one car travels towards the east and the other car travels towards the north. As the direction of two cars is different, the momentum of the two cars is also different.

From the analysis, it is clear that, the kinetic energy is the same for both the cars and momentum is different for each car.

Conclusion:

Therefore, the correct option is, “(b). They have the same kinetic energies, but different momenta”.