A cannon is fired from a cliff 190 m high downward at an angle of 21° with respect to the horizontal. If the muzzle velocity is 34 m/s, what is its speed (in m/s) when it hits the ground?
final KE= intial KE+initialPE
1/2 m v^2=1/2 m 34^2 + mg*190
solve for v
but i don't have m...
divide both sides by m. It divides out. You don't have to know the mass.
To find the speed of the cannon when it hits the ground, we need to analyze the motion of the cannonball in two dimensions: horizontally and vertically.
First, let's find the time it takes for the cannonball to hit the ground. We can use the vertical motion equation:
y = v₀yt + (1/2)gt²,
where
y = vertical displacement = -190 m (negative because the cannonball is moving downward),
v₀y = initial vertical velocity = v₀ * sin(θ),
g = acceleration due to gravity = 9.8 m/s² (assuming no air resistance),
t = time.
Rearranging the equation, we have:
-190 m = (v₀ * sin(21°))t - (1/2)(9.8 m/s²)t².
Solving this quadratic equation will give us the time it takes for the cannonball to hit the ground. Let's call the positive solution t₁.
Next, let's find the horizontal displacement of the cannonball using the horizontal motion equation:
x = v₀xt,
where
x = horizontal displacement,
v₀x = initial horizontal velocity = v₀ * cos(θ),
t = time. (We'll use t₁ in this case since it represents the total time of flight.)
Now that we have the horizontal displacement, we can calculate the speed of the cannonball when it hits the ground using the formula:
speed = distance / time.
In this case, the distance is the horizontal displacement x and the time is t₁.
Let's plug in the values and calculate the solution step by step:
1. Calculate the time it takes for the cannonball to hit the ground:
-190 = (34 m/s * sin(21°))t₁ - (1/2)(9.8 m/s²)t₁².
Solve this quadratic equation to find t₁.
2. Calculate the horizontal displacement:
x = (34 m/s * cos(21°)) * t₁.
3. Calculate the speed of the cannonball when it hits the ground:
speed = x / t₁.
By following these steps, you can find the speed at which the cannonball hits the ground.