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?
To find the speed of the cannon when it hits the ground, we can use the principles of projectile motion.
First, let's break down the initial velocity of the cannonball into its horizontal and vertical components.
The horizontal component of the velocity remains constant throughout the motion and is given by:
Vx = V * cos(θ)
where V is the magnitude of the initial velocity (34 m/s) and θ is the angle of projection (21°).
So, Vx = 34 * cos(21°).
Similarly, the vertical component of the velocity changes due to the acceleration due to gravity (9.8 m/s²). The equation to determine the vertical component of velocity as a function of time is:
Vy = V * sin(θ) - gt
where Vy is the vertical component of the velocity, V is the magnitude of the initial velocity (34 m/s), θ is the angle of projection (21°), g is the acceleration due to gravity (9.8 m/s²), and t is the time.
Since the cannonball hits the ground when it reaches the same vertical height as the initial position (190 m), we can use this information to determine the time it takes for the cannonball to hit the ground.
The equation for the vertical displacement as a function of time is given by:
y = V * sin(θ) * t - (1/2) * g * t²
where y is the vertical displacement (190 m), V is the magnitude of the initial velocity (34 m/s), θ is the angle of projection (21°), g is the acceleration due to gravity (9.8 m/s²), and t is the time.
Since we are trying to find the time it takes for the cannonball to hit the ground, we can set y equal to zero and solve for t:
0 = V * sin(θ) * t - (1/2) * g * t²
This is a quadratic equation in terms of t. Solving this equation will give us two values for t, one of which is negative (corresponding to when the cannonball is initially launched). We can disregard the negative value and consider the positive value as the time it takes for the cannonball to hit the ground.
Once we have the time t, we can use it to calculate the vertical component of the velocity when the cannonball hits the ground:
Vy = V * sin(θ) - g * t
Finally, we can calculate the speed of the cannonball when it hits the ground by using the Pythagorean theorem:
Speed = sqrt(Vx² + Vy²)