A billiard ball of mass m = 0.250 kg hits the cushion of a billiard table at an angle of θ1 = 56.2° and a speed of v1 = 24.8 m/s. It bounces off at an angle of θ2 = 71.0° and a speed of v2 = 10.0 m/s.
(a) What is the magnitude of the change in momentum of the billiard ball?
(b) In which direction does the change-of-momentum vector point? (Let up be the +y positive direction and to the right be the +x positive direction.)
Im confused on the momentum part becuase i know the change is momentum is momentum final subtracted by momentum initial but the angles are throwing off the outcome of my answer.
Just a quick question…
Did this problem come with a figure?
I can't tell if those angles are along the horizontal or vertical.
It does but i don't know how to get it in the box
To calculate the magnitude of the change in momentum (Δp), you can use the following formula:
Δp = √((px2 - px1)^2 + (py2 - py1)^2)
where px1 and py1 represent the initial x-component and y-component of momentum, and px2 and py2 represent the final x-component and y-component of momentum.
To find px1 and py1, use the initial velocity (v1) and the angle of incidence (θ1). The initial x-component of momentum (px1) can be found using the formula:
px1 = m * v1 * cos(θ1)
And the initial y-component of momentum (py1) can be found using:
py1 = m * v1 * sin(θ1)
Similarly, you can find px2 and py2 using the final velocity (v2) and the angle of reflection (θ2):
px2 = m * v2 * cos(θ2)
py2 = m * v2 * sin(θ2)
Once you have the values of px1, py1, px2, and py2, you can substitute them into the formula for Δp to find the magnitude of the change in momentum.
To determine the direction of the change-of-momentum vector, you can consider the sign of the changes in the x-component and y-component of momentum. If both components increase, the vector points in the positive direction of both axes. If both components decrease, the vector points in the negative direction of both axes. If one component increases and the other decreases, the vector points in the direction of the component that increases.
Note that in this problem, since the ball bounces off the cushion, the magnitude of the change in momentum will be the same as before the collision (due to the conservation of momentum). So, the magnitude of the change in momentum is equal to the momentum initial and the direction of the change-of-momentum vector will point in the same direction as the initial momentum.