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Emma (Parent) October 16 2016
This question involves the principle of the conservation of momentum.
Recall: Assuming no external forces act, linear momentum is always conserved. This means the total linear momentum of two objects before they collide equals the total linear momentum after the collision.
To start, let's work out the momentum of each ball separately. We'll call the moving ball "A" and the stationary ball "B". We will need the formula Momentum = Mass * Velocity, or p = m * v.
For ball A:
p = m * v
p = 0.25 * 1.2
p = 0.3 kg m s-1
Let's call this value pA.
For ball B:
p = m *v
p = 0.25 * 0
p = 0.
Let's call this value pB.
So now we can work out the total momentum before the collision:
ptotal = pA + pB = 0.3 + 0 = 0.3 kg m s-1.
This is where we bring in the principle of conservation of momentum. The question tells us that after the collision, the balls move together with the same velocity. This means we can treat them as one object, and use our formula to work out the velocity after the collision.
We know that after the collision, the total momentum = 0.3 kg m/s, from the principle of conservation.
We also know that the total mass of both balls (remember we're treating them as one object, so we can add the masses) is 0.5 kg.
We simply plug these numbers into our formula for momentum:
p = m * v
0.3 = 0.5 * v
v = 0.3/0.5
v = 0.6 m/ssee more
We are given the substitution to use, so the first step is to differentiate "u" with respect to x.
du/dx = -sin(x)
Now, to replace the "dx" in the original integrand with something in terms of "du", we rearrange the differential:
dx = -1/sin(x) du
We substitute this into the original expression we are integrating; this gives:
S -12sin(x)cos3(x) (-1/sin(x)) du
Let's do some simplifying here; the negative signs cancel, and so does sin(x):
S 12cos3(x) du
Now, simplify again using u=cos(x); this gives:
S 12u3 du
This is a simple C1-level integration; integrating with respect to "u" and adding a constant of integration, we get:
3u4 + c
For our final answer, replace "u" with cos(x):
3cos4x) + csee more