Two identical cannons fire cannonballs horizontally with the same initial velocity. One cannon is located on a platform partway up a tower, and the other is on top, four times higher than the first cannon. How much farther from the tower does the cannonball from the higher cannon land than the cannonball from the lower cannon?
they both go at the same horizontal speed forever and ever (until they hit)
Therefore we only have to solve a dropping problem
h and 4 h and we want the times in the air
h = (1/2) g t^2
4 h = (1/2) g T^2
t^2 = 2 h/g
T^2 = 8 h/g
T^2/t^2 = 4
so
T/t = 2
so the high one is in the air twice as long
so
It goes twice as far.
To solve this question, we need to analyze the motion of the cannonballs fired from the two cannons. Both cannonballs have the same initial velocity but are fired at different heights. Let's break down the problem step by step:
Step 1: Identify the key information given in the question.
- The cannons have identical initial velocities.
- The second cannon is located four times higher than the first cannon.
Step 2: Understand the motion of the cannonballs.
- Since the cannonballs are fired horizontally, they will travel only in the horizontal direction without any vertical acceleration.
- This means that the time of flight for both cannonballs will be the same since their vertical motions are not affected.
Step 3: Calculate the time of flight for the cannonballs.
- The time of flight can be determined by using the formula: time = distance / velocity.
- Since the initial velocities are the same for both cannonballs, this formula simplifies to time = distance.
- Since the cannonballs are fired horizontally, the time of flight will be the same for both.
Step 4: Find the distance traveled by each cannonball.
- The distance traveled by the cannonball from the lower cannon can be calculated by considering the horizontal velocity and time of flight.
- The distance traveled by the cannonball from the higher cannon will be the same, as they have the same time of flight.
- However, there will be an additional horizontal displacement for the higher cannonball due to its initial height.
Step 5: Calculate the additional horizontal displacement of the higher cannonball.
- The additional horizontal displacement is the difference between the horizontal distance traveled by the higher cannonball and the lower cannonball.
Given that the second cannon is four times higher than the first cannon, the higher cannonball will have an additional horizontal displacement of four times the distance traveled by the lower cannonball.
In summary, the cannonball from the higher cannon lands four times farther from the tower than the cannonball from the lower cannon.