a 1.5kg cannon is mounted on top of a 2.0kg cart and loaded with a 52.7 gram ball. The cannon, car, and ball are moving forward with a speed of 1.27m/s. The cannon is ignited and launches a 52.7 gram ball forward with a speed of 75m/s. Determine the post-explosion velocity of the cannon and cart.
m1 =1,5 kg, m2 = 2 kg, m3 =0.0527 kg.
v = 1.27 m/s
u1 =75 m/s,
u2 =?
The law of conservation of linear momentum
(m1+m2+m3) •v = (m1+m2) •u2 + m3•u1,
u2 = {(m1+m2+m3) •v - m3•u1}/(m1+m2) =
={(1.5+2+0.0527)1.27 – 0.0527•75}/(1.5+2)=
=(4.51-3.95)/3.5 = 0.16 m/s (in the previous direction)
Well, well, well, we've got ourselves a cannonball situation, don't we? Let's calculate the post-explosion velocity of the cannon and cart!
First things first, let's figure out the initial momentum before the explosion. The momentum of an object is given by the product of its mass and velocity. So, the initial momentum of the cannon and cart can be found by adding up the momentum of the cannon, cart, and ball.
The momentum of the cannon before the explosion is given by 1.5 kg (mass of the cannon) multiplied by 1.27 m/s (initial velocity of the cannon and cart).
The momentum of the cart can be found by multiplying its mass of 2.0 kg by the same initial velocity of 1.27 m/s.
And finally, the momentum of the ball before the explosion is given by 0.0527 kg (mass of the ball) multiplied by the initial velocity of 1.27 m/s.
Now, if we add up all these momenta, we'll have the initial momentum. But remember, momentum is a vector quantity, so we need to take into account the directions of motion. Assuming the forward direction is positive, the momentum of the cannon and cart is positive, the momentum of the ball is negative (opposite direction), and we'll consider up as positive as well.
After the explosion, the momentum of the cannon and cart is still conserved. So, the final momentum of the cannon and cart will be equal to the initial momentum. But this time, there's no ball involved, so we'll only have the final velocity of the cannon and cart.
Now, let's calculate! I hope you're ready for some cannonball math fun!
(1.5 kg x 1.27 m/s) + (2.0 kg x 1.27 m/s) + (-0.0527 kg x 1.27 m/s) = final velocity of the cannon and cart x (1.5 kg + 2.0 kg)
Solving for the final velocity of the cannon and cart, we get:
Final velocity of the cannon and cart = [(1.5 kg x 1.27 m/s) + (2.0 kg x 1.27 m/s) + (-0.0527 kg x 1.27 m/s)] / (1.5 kg + 2.0 kg)
Pop in the numbers, crunch them, and you'll have your post-explosion velocity of the cannon and cart!
To determine the post-explosion velocity of the cannon and cart, we can use the principle of conservation of momentum.
First, let's calculate the initial momentum of the system before the explosion:
Initial momentum = (Mass of Cannon + Mass of Cart + Mass of Ball) x Initial velocity
Mass of Cannon = 1.5 kg
Mass of Cart = 2.0 kg
Mass of Ball = 52.7 g = 0.0527 kg
Initial velocity = 1.27 m/s
Initial momentum = (1.5 kg + 2.0 kg + 0.0527 kg) x 1.27 m/s
Initial momentum = 3.5527 kg x 1.27 m/s
Initial momentum = 4.511 m/s
After the explosion, the cannon and cart will move in the opposite direction of the ball's momentum.
Final momentum = (Mass of Cannon + Mass of Cart) x Final velocity
Mass of Cannon = 1.5 kg
Mass of Cart = 2.0 kg
Final velocity = ?
Final momentum = (1.5 kg + 2.0 kg) x Final velocity
Final momentum = 3.5 kg x Final velocity
Since momentum is conserved, the initial momentum equals the final momentum:
Initial momentum = Final momentum
4.511 m/s = 3.5 kg x Final velocity
Now, we can solve for the final velocity of the cannon and cart:
Final velocity = 4.511 m/s / 3.5 kg
Final velocity ≈ 1.29 m/s
Therefore, the post-explosion velocity of the cannon and cart is approximately 1.29 m/s in the opposite direction of the ball's motion.
To determine the post-explosion velocity of the cannon and the cart, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
Before the explosion, the cannon, cart, and the ball all move together with a speed of 1.27 m/s. The total mass before the explosion is:
mass_before = mass_cannon + mass_cart + mass_ball
mass_before = 1.5 kg + 2.0 kg + 0.0527 kg
mass_before = 3.5527 kg
The total momentum before the explosion is given by:
momentum_before = mass_before * velocity_before
momentum_before = 3.5527 kg * 1.27 m/s
momentum_before = 4.5147 kg·m/s
After the explosion, the ball is launched forward with a speed of 75 m/s, while the cannon and cart move with an unknown velocity, which we need to find. Let's call this velocity "v_cannon-cart".
The momentum of the ball after the explosion is given by:
momentum_ball = mass_ball * velocity_ball
momentum_ball = 0.0527 kg * 75 m/s
momentum_ball = 3.9525 kg·m/s
The momentum of the cannon and cart after the explosion is given by:
momentum_cannon-cart = (mass_cannon + mass_cart) * v_cannon-cart
Now, applying the conservation of momentum, we can set up the equation:
momentum_before = momentum_ball + momentum_cannon-cart
4.5147 kg·m/s = 3.9525 kg·m/s + (1.5 kg + 2.0 kg) * v_cannon-cart
4.5147 kg·m/s = 3.9525 kg·m/s + 3.5 kg * v_cannon-cart
Now, we can solve this equation to find the post-explosion velocity of the cannon and cart:
v_cannon-cart = (4.5147 kg·m/s - 3.9525 kg·m/s) / (3.5 kg)
v_cannon-cart = 0.5622 kg·m/s / (3.5 kg)
v_cannon-cart = 0.1606 m/s
Therefore, the post-explosion velocity of the cannon and cart is approximately 0.161 m/s in the forward direction.