A colony of troglodytes has been in a lengthy feud with its neighbors on the adjacent cliff. Colony A finally develops an important military breakthrough: it rolls bombs off its cliff at known rates of speed, thus gaining pinpoint accuracy in its attacks.
If the cliffs are separated by 40.5 m and a bomb is rolled at 7.9 m/s, how far down the opposite cliff will it land? The acceleration due to gravity is 9.8 m/s^.
Answer in units of m.
horizontal speed is constant so
t = 40.5 / 7.9
so
distance down = (1/2) g t^2
= 4.9 t^2
Well, it seems that Colony A is quite literally "bombing" their way to victory! Now, let's calculate how far down the opposite cliff the bomb will land.
To find the horizontal distance traveled by the bomb, we'll need to determine the time it takes for the bomb to reach the opposite cliff. We can use the formula:
time = distance / speed
Plugging in the values, we get:
time = 40.5 m / 7.9 m/s
Calculating this, we find that it takes approximately 5.12658 seconds for the bomb to reach the opposite cliff.
Next, we'll need to determine how far the bomb falls vertically during this time. This can be calculated using the formula:
distance = (0.5) * acceleration * (time^2)
Plugging in the values, we get:
distance = (0.5) * 9.8 m/s^2 * (5.12658 s)^2
After crunching the numbers, we find that the bomb will fall approximately 126.58 meters down the opposite cliff.
So, to answer your question, the bomb will land about 126.58 meters down the opposite cliff. Just keep in mind that this is strictly for educational purposes and should not be taken as advice for settling any actual feuds!
To find the distance the bomb will travel, we can use the equations of motion under constant acceleration. The formula we can use is:
d = (v^2 - u^2) / (2a)
Where:
d = distance traveled
v = final velocity
u = initial velocity
a = acceleration
In this case, the bomb is rolled horizontally, so the initial vertical velocity (u) is 0 m/s. The acceleration (a) is due to gravity and is equal to -9.8 m/s^2 (negative because it acts in the opposite direction as positive vertical displacement). The final velocity (v) is the vertical velocity acquired by the bomb at the bottom of the cliff, which is what we want to find.
Using the given values:
u = 0 m/s
v = ?
a = -9.8 m/s^2
We can rearrange the equation and solve for v:
v^2 = 2ad
v^2 = 2 * (-9.8 m/s^2) * 40.5 m
v^2 = -2 * (-396.9 m^2/s^2)
v^2 = 793.8 m^2/s^2
Taking the square root of both sides, we get:
v ≈ 28.2 m/s
Now that we have the final vertical velocity, we can use it to find the distance the bomb will travel on the opposite cliff. Since the bomb is rolling horizontally, the time it takes to reach the bottom of the cliff will be the same as the time it takes to reach there vertically.
The time (t) can be found using the formula:
t = (v - u) / a
Substituting the known values:
t = (28.2 m/s - 0 m/s) / -9.8 m/s^2
t ≈ -2.9 s
Note: The negative sign in the time is due to the direction the acceleration is acting.
Finally, we can find the distance traveled horizontally:
d = v * t
d = 7.9 m/s * -2.9 s
d ≈ -22.91 m
Since distance cannot be negative in this case, we take the absolute value:
d ≈ 22.91 m
Therefore, the bomb will land approximately 22.91 meters down the opposite cliff.
To calculate the distance at which the bomb will land on the opposite cliff, we can use the equation of motion. Let's break down the problem step by step:
1. First, we need to determine the time it takes for the bomb to land. We can use the equation:
**d = v*t + (1/2)*a*t^2**
Where:
- d is the distance traveled (40.5 m in this case, as both cliffs are separated by 40.5 m)
- v is the initial velocity of the bomb (7.9 m/s)
- a is the acceleration due to gravity (-9.8 m/s^2 since it acts downward)
- t is the time taken
2. Rearranging the equation, we get:
**0 = (1/2)*(-9.8)*t^2 + 7.9*t - 40.5**
This equation is in the form of a quadratic equation, which can be solved using the quadratic formula:
**t = (-b ± √(b^2 - 4ac)) / (2a)**
Where:
- a = (1/2)*(-9.8) = -4.9
- b = 7.9
- c = -40.5
3. Substituting the values into the quadratic formula, we get:
**t = (-7.9 ± √(7.9^2 - 4*(-4.9)*(-40.5))) / (2*(-4.9))**
4. Simplifying the equation leads to two solutions for t:
t1 ≈ 1.30 seconds (positive solution)
t2 ≈ -6.31 seconds (negative solution)
We discard the negative solution as time cannot be negative in this scenario.
5. Once we have the time, we can calculate the distance the bomb will travel downward. Using the formula:
**d = v*t + (1/2)*a*t^2**
Substitute the values obtained:
- v = 0 m/s (since the bomb has zero initial velocity when it reaches the opposite cliff)
- a = -9.8 m/s^2 (acceleration due to gravity acting downward)
- t = 1.30 s
**d = 0*1.30 + (1/2)*(-9.8)*(1.30)^2**
6. Calculating the value gives:
**d ≈ -8.57 m**
However, since distance is always positive, we consider the absolute value:
**d ≈ 8.57 m**
Therefore, the bomb will land approximately 8.57 meters down the opposite cliff.
The troglodyte war continues, and a particularly offensive member of colony B is located 120 m below the top. At what speed must a bomb be rolled to get him?
Answer in units of m/s.