An object of mass 2kg moving with a speed 'u'collided with a stationary box , after collision , the object continue in its original direction with one quarter of its initial speed.
If the total kinetic energy of the colliding system remains constant , calculate the mass of the box .
Let the mass of the box be 'm'.
Before the collision, the kinetic energy of the system is:
K1 = 1/2 * 2kg * u^2 = u^2
After the collision, the object has one quarter of its initial speed, so its final speed is u/4. The box must have gained momentum in the opposite direction to conserve momentum.
Using conservation of momentum:
Initial momentum = Final momentum
2kg * u = 2kg * (u/4) + m * (u/4)
Simplifying:
u = (m/2 + 1) * (u/4)
4 = m/2 + 1
m/2 = 3
m = 6kg
Therefore, the mass of the box is 6kg.
To find the mass of the box, we will use the principle of conservation of kinetic energy.
Let's assume that the mass of the box is 'm' kg.
Before the collision, the object of mass 2kg was moving with speed 'u'. Therefore, its initial kinetic energy can be calculated as:
Initial kinetic energy = (1/2) * mass * velocity^2
= (1/2) * 2kg * u^2
= u^2
After the collision, the object continues in its original direction but with one quarter of its initial speed. So its final speed will be (u/4).
The final kinetic energy of the object can be calculated as:
Final kinetic energy = (1/2) * mass * velocity^2
= (1/2) * 2kg * (u/4)^2
= (1/2) * 2kg * (u^2/16)
= (u^2/16)
According to the principle of conservation of kinetic energy, the initial kinetic energy is equal to the final kinetic energy. Therefore, we have:
u^2 = (u^2/16)
Cross-multiplying, we get:
16u^2 = u^2
Simplifying, we get:
16u^2 - u^2 = 0
15u^2 = 0
Dividing both sides of the equation by 15, we get:
u^2 = 0
Since the value of 'u' is squared, it cannot be zero. Therefore, there is no solution to this equation.
This implies that the total kinetic energy of the colliding system cannot remain constant if the object continues with one quarter of its initial speed.
Hence, it is not possible to determine the mass of the box under the given conditions.