an 800 lb cannon fires a 5 lb cannonball in a horizontal direction. the cannon sits on a skid; the kinetic coefficient of friction between the skid and the horizontal surface is .25. if the cannonball has a speed of 650 ft/s when it leaves the cannon, determine
a) the recoil velocity of the cannon
b)the distance the cannon will slide before coming to rest
To determine the recoil velocity of the cannon, we can use the principle of conservation of momentum. The total initial momentum of the system (cannon and cannonball) is equal to the total final momentum.
a) Recoil velocity of the cannon (v_c):
The initial momentum of the system is given by the product of the cannon mass (m_c) and the initial velocity of the cannon (v_c_initial). The final momentum is given by the sum of the cannonball mass (m_b) and the cannonball's final velocity (v_b_final). Since the cannonball is fired horizontally, its final velocity is equal to its initial velocity.
The equation for conservation of momentum can be written as follows:
m_c * v_c_initial = (m_b * v_b_final) + (m_c * v_c_final)
Given:
m_c (mass of the cannon) = 800 lb
v_c_initial (initial velocity of the cannon) = 0 (since it is initially at rest)
m_b (mass of the cannonball) = 5 lb
v_b_final (velocity of the cannonball) = 650 ft/s
Substituting these values, we have:
800 lb * 0 = (5 lb * 650 ft/s) + (800 lb * v_c_final)
0 = 3250 lb * ft/s + 800 lb * v_c_final
Rearranging the equation to solve for v_c_final:
v_c_final = -3250 lb * ft/s / 800 lb
v_c_final ≈ -4.06 ft/s
Therefore, the recoil velocity of the cannon is approximately -4.06 ft/s. The negative sign indicates that the cannon recoils in the opposite direction of the cannonball's motion.
b) To determine the distance the cannon will slide before coming to rest, we can use the concept of work and energy. The work done against friction will be equal to the initial kinetic energy of the cannon.
The work done against friction (W_friction) is given by the equation:
W_friction = force of friction * distance
The force of friction can be calculated using the equation:
force of friction = coefficient of friction * normal force
The normal force acting on the cannon is equal to the weight of the cannon (800 lb) since it is resting on a horizontal surface. Therefore:
normal force = 800 lb
Substituting the given value of the coefficient of friction (0.25), we have:
force of friction = 0.25 * 800 lb
force of friction = 200 lb
The work done against friction is equal to the change in kinetic energy:
W_friction = ΔKE = KE_final - KE_initial
Since the cannon comes to rest (final velocity = 0), the final kinetic energy is 0. The initial kinetic energy (KE_initial) is given by:
KE_initial = (1/2) * m_c * v_c_initial^2
Substituting the given values, we have:
KE_initial = (1/2) * 800 lb * (4.06 ft/s)^2
KE_initial ≈ 1311.2 lb-ft^2/s^2
Therefore, the work done against friction (W_friction) is equal to 1311.2 lb-ft^2/s^2.
To find the distance the cannon will slide, we can rearrange the equation for work done against friction:
W_friction = force of friction * distance
distance = W_friction / force of friction
Substituting the values we have:
distance = 1311.2 lb-ft^2/s^2 / 200 lb
distance ≈ 6.56 ft
Therefore, the cannon will slide approximately 6.56 ft before coming to rest.