Chapter 8, Problem 18TP

To determine

(a)

An experimental procedure to determine the velocities of the carts before and after the collision and the equipment used in it.

Answer to Problem 18TP

The velocities of the carts before and after the collision can be determined by measuring the time taken by the carts to cover a distance.

Explanation of Solution

Introduction:

An experiment can be performed in which the two carts will be placed on a straight horizontal track. One of the carts will be kept at rest while the other will be pushed towards it to cause collision between the carts.

The additional equipment required to perform this experiment are a measuring tape to measure the distance travelled by the carts and a stopwatch to determine the time interval.

The expression for the average velocity in terms of distance and time is given as

  $v=\frac{d}{t}$

Measure the distance between the carts before pushing the first cart towards the second cart. Then push the first cart towards the second cart which is at rest and note the time taken by the first cart to hit the second cart using a stopwatch.

Substitute the obtained values in the above expression to calculate the average velocity of the first cart before the collision. The velocity of the second cart before the collision will be zero because it is at rest.

To calculate the velocities of the carts after the collision, repeat the same procedure. Measure the distance travelled by the carts and the time taken to travel the distance. After the collision, the carts get stick together and therefore, there velocities after the collision will be equal.

After the calculation of the velocities of the carts before and after the collision, the expression for the conservation of momentum can be used to determine the mass of the second cart.

  $m_{\text{A}}{v}_{\text{A}}+m_{\text{B}}{v}_{\text{B}}=m{v}^{\prime }$

Here, $m$ is the combined mass and $v^{\prime }$ is the combined velocity after the collision.

The combined mass is given by

  $m=m_{\text{A}}+m_{\text{B}}$

So, the expression to calculate the mass of cart $B$ is

  $m_{\text{B}}=\frac{m_{\text{A}}{v}_{\text{A}}+m_{\text{B}}{v}_{\text{B}}}{v^{\prime }}-m_{\text{A}}$

Conclusion:

The velocities of the carts before and after the collision can be determined by measuring the distance travelled by the carts and the time taken to cover the distance.

To determine

(b)

The factor which affects the uncertainty in the measurement of the mass of cart $B$ more.

Answer to Problem 18TP

The errors in the measurements taken before and after the collision will equally affect the uncertainty in the calculation of the mass of cart $B$.

Explanation of Solution

The mass of the cart $B$ is calculated by the following expression,

  $m_{\text{B}}=\frac{m_{\text{A}}{v}_{\text{A}}+m_{\text{B}}{v}_{\text{B}}}{v^{\prime }}-m_{\text{A}}$

The expression includes the values of the velocities of the carts calculated before the collision and the values of the velocities of the carts calculated after the collision. The velocities after and before the collision are calculated using same procedure. Therefore, the errors in the measurements taken before and after the collision will equally affect the uncertainty in the calculation of the mass of cart $B$.

Conclusion:

The errors in the measurements taken before and after the collision will equally affect the uncertainty in the calculation of the mass of cart $B$.

To determine

(c)

Themass of cart $B$.

Answer to Problem 18TP

The mass of cart $B$ is $0.75\text{kg}$.

Explanation of Solution

Given:

The mass of cart $A$ is $m_{\text{A}}=0.50\text{kg}$.

Formula used:

The conservation of momentum for combined bodies is given by

  $m_1{v}_1+m_2{v}_2=m{v}^{\prime }$

Here, $m$ is the combined mass and $v^{\prime }$ is the combined velocity after the collision.

The combined mass is given by

  $m=m_1+m_2$

The velocity is given by

  $v=\frac{d}{t}$

Calculation:

The initial velocity of cart $A$ is calculated as

  $\begin{array}{c}{v}_{\text{A}}=\frac{d}{t}\\ =\frac{1.5\text{m}}{1\text{s}}\\ =1.5\text{m}/\text{s}\end{array}$

The combined velocity after the collision is calculated as

  $\begin{array}{c}{v}^{\prime }=\frac{d}{t}\\ =\frac{0.6\text{m}}{1\text{s}}\\ =0.6\text{m}/\text{s}\end{array}$

The mass of the cart $B$ is calculated as

  $\begin{array}{c}{m}_{\text{B}}=\frac{m_{\text{A}}{v}_{\text{A}}+m_{\text{B}}{v}_{\text{B}}}{v^{\prime }}-m_{\text{A}}\\ =\frac{\left(0.50\text{kg}\right)\left(1.5\text{m}/\text{s}\right)+m_{\text{B}}\left(0\text{m}/\text{s}\right)}{0.6\text{m}/\text{s}}-0.50\text{kg}\\ =1.25\text{kg}-0.50\text{kg}\\ =0.75\text{kg}\end{array}$

Conclusion:

The mass of cart $B$ is equal to $0.75\text{kg}$.

To determine

(d)

Whether thevalues of initially measured physical quantities affect the loss in energy during inelastic collision.

Explanation of Solution

Introduction:

In an inelastic collision, a part of kinetic energy gets lost into the other forms of energy after the collision. Therefore, the kinetic energy does not remain conserved in an inelastic collision. The amount of energy lost after the collision can be simply determined by measuring the difference in the kinetic energy of the body before and after the collision. To measure the kinetic energy of a body, it is required to have the values of the mass and the velocity of the body.

The expression for the energy losses during the inelastic collision is given by

  $f_{\text{loss}}=\left(\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2\right)-\left(\frac{1}{2}{m}_1{v^{\prime }}_1^2+\frac{1}{2}{m}_2{v^{\prime }}_2^2\right)$

It can be seen from the above expression that the values of the velocities of both the carts before the collision affects the magnitude of energy losses during the collision.

Conclusion:

The initially measured physical quantities will affect the amount of energy lost during the inelastic collision on the basis of the conservation of the kinetic energy.