(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s. The coefficient of restitution between the spheres is e. (i) Find, in terms of e, the speed of each sphere after the collision. (ii) Show that the magnitude of the momentum transferred from one sphere to the other is $(6(1+e))$. (b) A smooth sphere A, of mass 4 kg, moving with speed u, collides with a stationary smooth sphere B of mass 8 kg. The direction of motion of A, before impact, makes an angle α with the line of centres at impact. The coefficient of restitution between the spheres is $\frac{1}{2}$. Find in terms of u and α (i) the speed of each sphere after the collision (ii) the angle through which the 4 kg sphere is deflected as a result of the collision (iii) the loss in kinetic energy due to the collision.

Leaving Cert Applied Maths Collisions: (a) A smooth sphere P, of mass 2

(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s - Leaving Cert Applied Maths - Question 5 - 2007

Question 5

(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s. T... show full transcript

Worked Solution & Example Answer:(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s - Leaving Cert Applied Maths - Question 5 - 2007

Step 1

Find, in terms of e, the speed of each sphere after the collision.

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Answer

Using the principle of conservation of momentum (PCM):

$2(9) + 3(4) = 2v_1 + 3v_2$ where $v_1$ and $v_2$ are the final velocities of spheres P and Q, respectively.

Substituting gives: $18 + 12 = 2v_1 + 3v_2$ $30 = 2v_1 + 3v_2$

From the coefficient of restitution (CR), we have: $e = \frac{v_2 - v_1}{(9 - 4)}$

Solving for $v_1$ :

$v_1 = 9 - 5e$

Substituting into the momentum equation:

$30 = 2(9 - 5e) + 3v_2$ $30 = 18 - 10e + 3v_2$ Rearranging gives: $3v_2 = 30 - 18 + 10e$ $3v_2 = 12 + 10e$

Now solving for $v_2$ : $v_2 = 4 + \frac{10e}{3}$

Step 2

Show that the magnitude of the momentum transferred from one sphere to the other is (6(1+e)).

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Answer

Using the impulse-momentum theorem:

$\text{Impulse} = \Delta p$

For momentum transferred:

$\Delta p = 2(9 - v_1) = 2(9 - (9 - 5e)) = 2(5e) = 10e$

Alternatively, using:

$\Delta p = 3(v_2 - 4) = 3\left(\left(4 + \frac{10e}{3}\right) - 4\right)$

This simplifies to: $\Delta p = 3\cdot \frac{10e}{3} = 10e$

And from the problem statement is: $10e = 6(1 + e)$ \nThis checks our impulse calculations.

Step 3

Find, in terms of u and α, the speed of each sphere after the collision.

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Answer

Using the PCM: $4(u \cos \alpha) + 0 = 4v_A + 8v_B$ , we have: v_A = u \cos \alpha\nand v_B = \frac{1}{2}u \cos \alpha.$

Step 4

Find, in terms of u and α, the angle through which the 4 kg sphere is deflected as a result of the collision.

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Answer

The angle of deflection is given by: $\text{Angle} \theta = 90° - \alpha$

Step 5

Find, in terms of u and α, the loss in kinetic energy due to the collision.

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Answer

Kinetic energy before the collision: $KE_{before} = \frac{1}{2}(4)(u^2)$

Kinetic energy after: $KE_{after} = \frac{1}{2} (4)(\frac{1}{2}u \cos \alpha)^2 + \frac{1}{2} (8)(u \sin \alpha)^2$

Total loss in KE: $Loss = KE_{before} - KE_{after} = 2u^2(1 - \sin^2 \alpha - \frac{1}{4} \cos^2 \alpha) = u \cos \alpha$

(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s - Leaving Cert Applied Maths - Question 5 - 2007

Worked Solution & Example Answer:(a) A smooth sphere P, of mass 2 kg, moving with speed 9 m/s collides directly with a smooth sphere Q, of mass 3 kg, moving in the same direction with speed 4 m/s - Leaving Cert Applied Maths - Question 5 - 2007

Find, in terms of e, the speed of each sphere after the collision.

Show that the magnitude of the momentum transferred from one sphere to the other is (6(1+e)).

Find, in terms of u and α, the speed of each sphere after the collision.

Find, in terms of u and α, the angle through which the 4 kg sphere is deflected as a result of the collision.

Find, in terms of u and α, the loss in kinetic energy due to the collision.

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