Theory question 118
Vectors(a) The position vectors of the points X and Y are x = (−2i + 5j) and y = (i − 7j) respectively. Find (i) (3x + 2y); (ii) (y − 2x); (iii) the angle between x and y; (iv) the unit vector in the direction of (x + y).
(b) A bullet of mass 0.084kg is fired horizontally into a 20 kg block of wood at rest on a smooth floor. If they both move at a velocity of 0.24ms−1 after impact, calculate, correct to two decimal places, the initial velocity of the bullet.
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(a)(i) 3x + 2y = (−6i + 15j) + (2i − 14j) = −4i + j
(ii) y − 2x = (i − 7j) − (−4i + 10j) = 5i − 17j
(iii) x·y = −2 − 35 = −37; |x| = √29, |y| = √50; cos θ = −37/√1450 ≈ −0.9717, θ ≈ 166.3°
(iv) x + y = −i − 2j, |x + y| = √5; unit vector = (−i − 2j)/√5
(b) Momentum: 0.084u = (20 + 0.084)(0.24) = 4.82016
u ≈ 57.38 ms−1
Use the dot product for the angle and conservation of momentum for the bullet.