A cannon shoots a shell straight up. It reaches its maximum height, 1,051 feet, and splits into two pieces, one weighing 2 lb and the other 4 lb. The two pieces are observed to strike the ground simultaneously. The 4 lb piece hist the ground 1,608 feet away from the explosion (measured along the x axis). How long would it have taken the shell to return to the ground if it has not split? (answer to two decimal points)

Question

A cannon shoots a shell straight up. It reaches its maximum height, 1,051 feet, and splits into two pieces, one weighing 2 lb and the other 4 lb. The two pieces are observed to strike the ground simultaneously. The 4 lb piece hist the ground 1,608 feet away from the explosion (measured along the x axis). How long would it have taken the shell to return to the ground if it has not split? (answer to two decimal points)

Answer 1

From the maximum height reached, you can determine the launch velocity

V = sqrt (2 g H) = 260 ft/s. Since the pieces at the ground arrive at the same time, they split in such a way that they acquire no vertical velocity components; otherwise one would go up and the other down. The time required for the pieces to fall after separating is T = sqrt (2*1051 ft/g) = 8.08 s

Since the the separation explosion occurred at the top of the trajectory and did not make the fall time different for the two pieces, the time to fall without the explosion would be the same as observed, 8.08 s. The info about how far away one piece landed is not needed.

Answer 2

OK thanks. There is another part of the question that asks for the momentum of the 4 lb piece and I can't figure out how to solve that.

Answer 3

To solve this problem, we need to use the laws of projectile motion. We can assume that the cannonball's motion follows a parabolic path, neglecting air resistance.

Let's denote the time it takes for the 2 lb portion to hit the ground as t1 and the time it takes for the 4 lb portion to hit the ground as t2.

Since both pieces hit the ground simultaneously, we can equate the times:

t1 = t2

Now, let's calculate the time it takes for the 4 lb piece to reach the ground.

Using the equation of motion for vertical displacement:

h = v0 * t - (1/2) * g * t^2

where:
h is the height (maximum height of 1,051 feet)
v0 is the initial vertical velocity (which is zero at the maximum height)
g is the acceleration due to gravity (32.2 ft/s^2)

Plugging in the values:

1,051 = 0 * t - (1/2) * 32.2 * t^2
1,051 = -16.1 * t^2

Rearranging the equation:

t^2 = -1,051 / -16.1
t^2 = 65.28
t = √65.28
t = 8.09 seconds

The time it took for the 4 lb piece to reach the ground is approximately 8.09 seconds. Since the 2 lb piece and the shell follow the same path up until the split, we know that it would also take approximately 8.09 seconds for the shell to return to the ground if it had not split.

Therefore, the answer is 8.09 seconds (rounded to two decimal points).