Understood, let's proceed without LaTeX.
(a) To find the impulse received by the ball when it strikes the wall, we use the impulse-momentum theorem:
Change in momentum equals final momentum minus initial momentum.
Given:
- Initial velocity = 8i m/s
- Final velocity = λ(i + 2j) m/s
- Mass = 1.5 kg
Change in momentum = (1.5 * λ(i + 2j)) - (1.5 * 8i)
= 1.5λ(i + 2j) - 12i
So, the impulse received by the ball is 1.5λ(i + 2j) - 12i.
(b) To find the value of the coefficient of restitution (e), we use the formula:
e equals relative velocity after impact divided by relative velocity before impact.
Relative velocity after impact equals final velocity minus wall velocity, which equals λ(i + 2j) - 0 (since the wall is fixed).
Relative velocity before impact equals initial velocity minus wall velocity, which equals 8i - 0 (since the wall is fixed).
So, e equals λ(i + 2j) divided by 8i.
Now, the kinetic energy lost by the ball when it strikes the wall is given as 33 J. We know that the kinetic energy lost during the collision is related to the coefficient of restitution by the equation:
Kinetic energy lost equals half the mass times the initial velocity squared times (1 - e squared).
Given kinetic energy lost = 33 J, mass = 1.5 kg, and initial velocity = 8 m/s.
Now, we can solve this equation to find the value of e.
Given:
- Kinetic energy lost = 33 J
- Mass = 1.5 kg
- Initial velocity = 8 m/s
We know that the kinetic energy lost during the collision is related to the coefficient of restitution (e) by the equation:
Kinetic energy lost = 0.5 * mass * (initial velocity)^2 * (1 - e^2)
Substituting the given values:
33 = 0.5 * 1.5 * 64 * (1 - e^2)
33 = 48 * (1 - e^2)
Now, we can solve for e:
Divide both sides by 48:
33 / 48 = 1 - e^2
11 / 16 = 1 - e^2
Subtract 1 from both sides:
-5 / 16 = -e^2
Divide both sides by -1:
5 / 16 = e^2
Take the square root of both sides:
e = sqrt(5 / 16)
e = sqrt(5) / 4
So, the value of e is sqrt(5) / 4.
