Q29E - University Physics with Modern Physics

Question

A little red wagon with mass 7.00 kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4 m/s and then is pushed 3.0 m in the direction of the initial velocity by a force with a magnitude of 10.0 N. (a) Use the work energy theorem to calculate the wagon’s final speed.

(b) Calculate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagon’s final speed. Compare this result to that calculated in part (a)

Step-by-Step Solution

Answer

a) The final speed of the wagon is 4.96 m/s.

b) The acceleration of the wagon due to force is 1.43 $m/s^2$.

Step 1: Identification of given data

The given data can be listed below,

• The mass of the red wagon is, m = 7.0 kg
• The initial speed of the wagon is, $v_i=4\hspace{0.33em}m/s$
• The distance wagon is pushed, s = 3 m
• The force on the wagon is, F = 10.0 N

Step 2: Concept/Significance of work energy theorem

The net work performed on a particle and that particle's velocity are directly related by the work-energy equation.

Step 3: (a) Determination of the wagon’s final speed

According to the work energy theorem, the final velocity of the wagon is given by,

 

$$\begin{gathered} W=K_f-K_i \\ =\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2 \\ {v}_f^2=\frac{2\left(W+\frac{1}{2}m{v}_i^2\right)}{m} \end{gathered}$$                                                                                                 …(i)

 

The work done is calculated by,

 

                                        W = Fs

 

Here, F is the force acting on the wagon, and s is the distance traveled by the wagon when it is pulled.

 

Substitute all the values in the above,

 

W = 10 N(3 m)

    =30 J

 

Substitute all the values in equation (i),

 

$$\begin{gathered} v_f=\sqrt{\frac{2\left(W+{\displaystyle \frac{1}{2}}m{v}_i^2\right)}{m}} \\ =\sqrt{\frac{2\left(30 J+0.5\times \left(7 kg\right){\left(4 m/s\right)}^2\right)}{7 kg}} \\ =\sqrt{\frac{2\left(30+56\right) J}{7 kg}} \\ =4.96 m/s \end{gathered}$$ 

 

Thus, the final speed of the wagon is 4.96 m/s.

Step 4: (b) Determination of the acceleration produced by the force

The acceleration of the wagon is given by,

 

$a=\frac{F}{m}$ 

 

Here, F is the force on the wagon, and m is the mass of the wagon.

 

Substitute all the values in the above,

 

 $$\begin{gathered} a=\frac{10 N}{7 kg} \\ =1.43 m/s^2 \end{gathered}$$

 

Thus, the acceleration of the wagon due to force is $1.43 m/s^2$.