projectile motion with constant horizontal speed ? A stone falls off a cliff with horizontal velocity v m/s. it falls a distance of 5cm vertically and 200m horizontally in t seconds. when it has fallen a vertical distance of 20cm, what is its horizontal distance from the cliff? (its horizontal speed is constant) how to solve this problem using proportion?
isnt horizontal distance proportional to time (yes).
isn't vetical distance proportional to time squared?
t1/t2=H1/H2= sqrt(V1/V2)
so V1=5cm, V2=20 cm so sqrt (V1/V2)=1/2
so horizontal distance H2=2*200m
To solve this problem using proportions, we first need to understand the concept of projectile motion with constant horizontal speed.
In projectile motion, an object moves in a curved path under the influence of gravity. When the horizontal velocity is constant, it means that the object's speed remains the same throughout its motion horizontally. However, vertically, the object accelerates due to the force of gravity.
Let's go step by step to solve the problem using proportions:
1. Start by identifying the given information:
- Vertical distance fallen: 5 cm
- Horizontal distance: 200 m
- Time: t seconds
- Vertical distance at a certain point: 20 cm
2. Use the formula for the vertical motion of an object under gravity:
h = ut + (1/2)gt^2
where h is the vertical displacement, u is the initial vertical velocity (which is 0 in this case since the stone falls off a cliff), g is the acceleration due to gravity.
Plugging in the given information, we have:
5/100 = (1/2)g(t^2) --> [Equation 1]
3. Now, let's find the value of g in terms of t using the distance formula:
g = 2h/t^2
Plugging in the value of h as 5/100, we get:
g = 2(5/100)/t^2
= 0.1/t^2
4. Next, we need to determine the time it takes for the stone to fall a vertical distance of 20 cm. We can use Equation 1 again with the given vertical distance, h = 20 cm.
20/100 = (1/2)g(t^2) --> [Equation 2]
5. Substitute the value of g from step 3 into Equation 2:
20/100 = (1/2)(0.1/t^2)(t^2)
= 0.1/2
= 0.05
6. Solve for t in Equation 2:
t = sqrt((20/100) / (0.05))
7. Calculate the value of t.
8. Once you have the value of t, you can find the horizontal distance using the formula:
horizontal distance = horizontal speed * time
Since the horizontal speed is given as v m/s, and the time is the value from step 7, you can calculate the horizontal distance.
By following these steps and using proportions, you can solve the problem.