A cannon is placed on a hillside pointing up the hill, where the hill is at an angle �a Pi/4 to the horizontal.
The cannon points at an angle �Pi/6 to the slope.
and a shot is �red uphill with initial speed v0.
(i) Find the maximum value of the distance between the cannon ball and the slope in the direction normal to the slope.
(ii) Find where the ball lands.
To find the maximum value of the distance between the cannonball and the slope in the direction normal to the slope, we need to analyze the projectile motion of the cannonball and determine its trajectory.
(i) First, let's analyze the horizontal and vertical components of the initial velocity of the cannonball.
The horizontal component of the initial velocity (v0x) can be determined using the cosine function:
v0x = v0 * cos(Pi/6)
The vertical component of the initial velocity (v0y) can be determined using the sine function:
v0y = v0 * sin(Pi/6)
Next, we can analyze the vertical motion of the cannonball. The vertical motion is influenced by gravity, which acts in the downward direction.
Using the equations of motion, we can determine the time (t) it takes for the cannonball to reach the maximum height:
v0y = u + at
Since the ball reaches the maximum height when it stops moving upward, the final vertical velocity (vf) would be zero. The initial vertical velocity (v0y) is given, and the acceleration due to gravity (g) is acting in the opposite direction. Hence:
0 = v0y - g * t
Solving this equation for t, we get:
t = v0y / g
Now, we can determine the maximum height (h) attained by the cannonball. To find this, we can use the equation of motion:
s = ut + (1/2) * a * t^2
Since the final displacement (s) at the maximum height is zero, the initial displacement (u) is zero, and the acceleration (a) is -g. Hence, we have:
0 = 0 + (1/2) * (-g) * t^2
Simplifying this equation, we get:
t^2 = 2 * h / g
Solving for h, we obtain:
h = (v0y)^2 / (2 * g)
The maximum value of the distance between the cannonball and the slope in the direction normal to the slope is equal to the maximum height attained by the cannonball (h).
(ii) To find where the ball lands, we need to consider the horizontal motion of the cannonball.
The horizontal distance (D) traveled by the cannonball can be determined using the equation of motion:
D = v0x * t
We already know the value of v0x. Using the previously derived value of t, we can substitute it into the equation to find the horizontal distance traveled by the ball.
D = (v0 * cos(Pi/6)) * (v0y / g)
Simplifying this expression, we obtain:
D = (v0^2 * cos(Pi/6) * sin(Pi/6)) / g
Thus, the horizontal distance D gives us the location where the cannonball lands.
By calculating these values, you would be able to find the answers to both (i) and (ii).