Suppose a ball of putty moving horizontally with 1kg.m/s of momentum collides and sticks to an identical ball of puffy moving vertically with 1 kg.m/s of momentum. Why is their combined momentum not simply the arithmetic sum, 2 kg.m/s?
Vector sum of two vectors at right angles
p=sqrt{p₁²+p₂²} =sqrt2 =1.41 kg•m/s
The combined momentum of the two balls of putty after the collision is not simply the arithmetic sum because momentum is a vector quantity. It takes into account both the magnitude and direction of the momentum.
In this case, the ball of putty moving horizontally has momentum in the x-direction, while the ball of putty moving vertically has momentum in the y-direction. When they collide and stick together, the resulting combined momentum will be the vector sum of their individual momenta.
Using vector addition, the combined momentum can be calculated using the Pythagorean theorem. The magnitude of the combined momentum can be found by taking the square root of the sum of the squares of the individual momenta. In this case, since both balls have equal momentum, the magnitude of the combined momentum will be √(1^2 + 1^2) = √(2) = √2 kg.m/s.
The direction of the combined momentum can also be calculated using trigonometry. Since one ball was moving horizontally and the other vertically, the resulting combined momentum will be at an angle of 45 degrees with respect to both the x and y-axes.
The combined momentum of the two balls after the collision is not simply the arithmetic sum of their individual momenta because momentum is a vector quantity. In the case of a one-dimensional collision like this, where the balls are moving in different directions, we need to consider the direction of the momentum as well.
When two objects collide and stick together, their momenta are added together, taking into account their direction. The momentum of an object moving to the right is considered positive, while the momentum of an object moving to the left is considered negative.
In this scenario, the putty ball moving horizontally has a momentum of 1 kg.m/s to the right, and the puffy ball moving vertically has a momentum of 1 kg.m/s upwards. Since the upward direction is usually considered positive, we can assign the vertical momentum a positive value.
When the two balls collide and stick together, their momentum vectors combine. The horizontal momentum of the putty ball is added to the vertical momentum of the puffy ball. Since these two vectors are perpendicular to each other, they cannot be added together as simple arithmetic sums.
Using vector addition, we can calculate the combined momentum using the Pythagorean theorem. The magnitude of the combined momentum is the square root of the sum of the squares of the horizontal and vertical momenta:
Combined Momentum = √(Horizontal Momentum^2 + Vertical Momentum^2)
In this case, the magnitude of the combined momentum would be √(1^2 + 1^2) = √2 ≈ 1.41 kg.m/s.
Therefore, the combined momentum of the two balls after the collision is approximately 1.41 kg.m/s, not simply 2 kg.m/s.