A cannon tilted up at a 26.0 degree angle fires a cannon ball at 78.0 m/s from atop a 12.0 m-high fortress wall. What is the ball's impact speed on the ground below?
u = constant horizontal speed = 78 cos 26
Vi = initial up speed = 78 sin 26
v = Vi - g t
h = 12 + Vi t - 4.9 t^2
at ground, h = 0
4.9 t^2 -Vi t - 12 = 0
solve for t, time in air
then use that t so solve for v at t in v=Vi -9.8 t
then impact speed = sqrt(u^2+v^2)
To solve this problem, we can break it down into two parts: finding the time of flight and then using that time to calculate the horizontal distance traveled by the cannonball. Finally, we can use this information to determine the impact speed.
First, let's find the time of flight. Since the initial velocity of the cannonball is launched at an angle, we need to separate it into its horizontal and vertical components.
Horizontal component (Vx):
Vx = V * cos(θ)
Vx = 78.0 m/s * cos(26.0°)
Vx = 70.33 m/s
Vertical component (Vy):
Vy = V * sin(θ)
Vy = 78.0 m/s * sin(26.0°)
Vy = 33.87 m/s
Next, we can use the vertical component (Vy) to calculate the time of flight (T). We'll use the kinematic equation for vertical motion:
Vertical displacement (Δy) = Vy * T + (1/2) * g * T^2
Since the cannonball is fired from atop a 12.0 m-high fortress wall, the vertical displacement is -12.0 m (negative because the cannonball is moving downwards).
-12.0 m = 33.87 m/s * T + (1/2) * (-9.8 m/s^2) * T^2
Rearranging the equation gives us a quadratic equation:
-4.9 T^2 + 33.87 T - 12.0 = 0
Solving this quadratic equation will give us the time of flight (T).
Using the quadratic formula:
T = (-b ± √(b^2 - 4ac)) / (2a)
where a = -4.9, b = 33.87, and c = -12.0
T = (-33.87 ± √(33.87^2 - 4 * -4.9 * -12.0)) / (2 * -4.9)
Solving this equation will give two values for T, one of which will be negative because it corresponds to the ball's initial ascent. We only consider the positive value, which represents the time taken to reach the ground.
After finding the positive value for T, we can use it to calculate the horizontal distance traveled by the cannonball.
Horizontal distance (d):
d = Vx * T
Finally, we can determine the impact speed on the ground below by calculating the resultant velocity just before impact. We can use the Pythagorean theorem:
Impact speed = √(Vx^2 + Vy^2)
Substituting the known values into the equation will give us the impact speed of the cannonball on the ground below.