A body of mass 2 kg moving with a velocity of $(\hat{\mathrm{i}}+$ $2\hat{\mathrm{j}}-$ $3\hat{\mathrm{k}})$ ms${}^{-1}$ collides with another body of mass 3 kg moving with a velocity of $(2\hat{\mathrm{i}}+$ $\hat{\mathrm{j}}+$ $\hat{\mathrm{k}})$ ms${}^{-1}$ . If they stick together, the velocity in ms${}^{-1}$ of the composite body will be:

1. $\frac{1}{5}(8\hat{i}+$ $7\hat{j}-$ $3\hat{k})$

2. $\frac{1}{5}(-4\hat{i}+$ $\hat{j}-$ $3\hat{k})$

3. $\frac{1}{5}(8\hat{i}+\hspace{0.17em}\hat{\mathrm{j}}-$ $\hat{\mathrm{k}})$

4. $\frac{1}{5}(-4\hat{i}+$ $7\hat{j}-$ $3\hat{k})$

If a 50 kg mass is swinging in a vertical plane on a string at rest then the power delivered by gravity when the mass is moving with a velocity of 2 m/sec upwards in a direction, making an angle of $60°$ with the vertical will be: (g = 9.8 m/$s^2$)

A force of 5 N making an angle $\theta$ with the horizontal acting on an object displaces it by 0.4 m along the horizontal direction. If the object gains kinetic energy of 1 J then the component of the force is:

A flexible smooth track is fixed in two alternate arrangements, as shown in figures 1 and 2. The length of the track used is the same in each case, and the height through which it falls from the bench to the floor is the same. A toy car is released at rest and slides down the track (One after the other on both the tracks). Air resistance can be ignored. Which of the following statement is true?
            
1. The speed at the bottom, as well as the time taken on both the tracks, are the same.

2. The speed at the bottom, as well as the time taken on both the tracks, are different.

3. The speed at the bottom is different but the time taken on both the tracks is the same.

4. The speed at the bottom is the same but the time taken on both the tracks is different.

A truck of mass \(30,000~\text{kg}\) moves up an inclined plane of slope \(1\) in \(100\) \(\left(\tan\theta = \frac{1}{100}\right)\) at a speed of \(30~\text{km/h}\). The power of the truck is: (given \(g=10~\text{ms}^{-2}\)):

A pump ejects 12,000 kg of water at a speed of 4 m/s in 40 s. The average rate at which the pump is working is:

1.  0.24 kW

2.  2.4 W

3.  2.4 kW

4.  24 W

A spring 40 mm long is stretched by the application of force. If 10 N force is required to stretch the spring through 1 mm, then work done to stretch the spring 40 mm is equal to:

A block of mass 2 kg moving with a velocity of 10 m/s on a smooth surface hits a spring of force constant $80\times {10}^3$ N/m as shown. The maximum compression in the spring will be:
               

A body is displaced from (0, 0) to (1m, 1m) along the path x=y by a force $F=\left(x^2\hat{j}+y\hat{i}\right)N$. The work done by this force will be :

1. $\frac{4}{3}J$

2. $\frac{5}{6}J$

3. $\frac{3}{2}J$

4. $\frac{7}{5}J$

A block of mass m initially at rest, is dropped from a height h onto a spring of force constant k. If the maximum compression in the spring is x, then:     
         

1. $mgh=\frac{1}{2}k{x}^2$

2. $mg(h+x)=\frac{1}{2}k{x}^2$

3. $mgh=\frac{1}{2}k{(x+h)}^2$

4. $mg(h+x)=\frac{1}{2}k{(x+h)}^2$