Question

A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant k = 2.05 ✕ 10⁴ N/m, as shown in the figure below. The cannon fires a 200 kg projectile at a velocity of 125 m/s directed 45.0° above the horizontal.

An illustration shows a carriage whose wheels are moving along horizontal rails. The carriage is horizontal to the rail. A cannon is placed on the carriage, and the barrel of the cannon is at an angle of 45° to the horizontal. A diagonal arrow parallel to the direction of the barrel extends from the barrel up and to the right. The right side of the horizontal base of the carriage is attached to a horizontal spring, which is connected at the other end to a vertical post.

(a) If the mass of the cannon and its carriage is 5690 kg, find the recoil speed of the cannon.

(I got 3.1067 and it was wrong.)

(b) Determine the maximum extension of the spring.

(c) Find the maximum force the spring exerts on the carriage.

Answer 1

PART (a):

Apply law of conservation of momentum (horizontal):

(200) x (125.cos 45^o) = (5690 kg) x (V)

V = ((200) x (125.cos 45^o)) / (5690 kg)

V = 3.107 m/s ----------------- (**Answer)

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PART (b):

Apply law of conservation of energy:

1/2 mv² = 1/2 kx²

x² = (mv²) / (k)

x² = (5690 kg) x (3.107 m/s)² / (2.05 x 10⁴ N/m)

x² = 2.679 m²

x = 1.637 m  ----------------- (**Answer)

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PART (c):

Maximum force is given by:

F_max = Kx

F_max = (2.05 x 10⁴ N/m) x (1.637 m)

F_max = 33554.1 N ----------------- (**Answer)