2. A bullet is fired from the ground at an angle of 45o above the horizontal. What initial speed vo must the bullet have in order to hit a point 550 ft high on a tower located 600 ft away (ignoring air resistance)?
Write separate equations for x and y with Vo and t as the unknowns. Require that x = 600 ft and y = 550 ft at the same time t. Solve for Vo and t.
To find the initial speed (vo) of the bullet, we can use the equations of motion for projectile motion.
First, we need to break down the initial velocity into its horizontal and vertical components. Since the bullet is fired at an angle of 45 degrees above the horizontal, these components can be determined using trigonometry.
The horizontal component of the initial velocity (vxo) remains constant throughout the motion and is given by:
vxo = vo * cosθ
where vo is the initial speed and θ is the angle.
The vertical component of the initial velocity (vyo) can be calculated as:
vyo = vo * sinθ
Using these components, we can analyze the vertical motion of the bullet. The vertical distance covered by the bullet can be expressed as:
y = y0 + vyo * t - (1/2) * g * t^2
where y is the vertical distance traveled (550 ft), y0 is the initial vertical position (0 ft), vyo is the vertical component of the initial velocity, t is the time of flight, and g is the acceleration due to gravity (32.2 ft/s^2).
In projectile motion, the time of flight is the time taken for the bullet to reach the ground. At the highest point (when the bullet hits the tower), the vertical velocity component becomes zero. Therefore, using the equation vyo - g * t = 0, we can find the time of flight (t) as:
t = vyo / g
By substituting this value of t back into the equation for y, we can solve for vo.
550 ft = 0 + vyo * (vyo / g) - (1/2) * g * (vyo / g)^2
Simplifying the equation will give us a quadratic equation in terms of vyo. Solving for vyo will give us the vertical component of the initial velocity.
Once vyo is known, we can use the equation for vxo to find the horizontal component:
vxo = vo * cosθ.
Finally, we can calculate the initial velocity (vo) by using the Pythagorean theorem:
vo = √(vxo^2 + vyo^2).
By following these steps and plugging in the given values (θ = 45 degrees, y = 550 ft, y0 = 0 ft, g = 32.2 ft/s^2), we can calculate the initial speed (vo) of the bullet.