Consider a projectile fired horizontally from a cliff of a given height. With what speed must it be fired so that it makes a 45 degree angle with the ground when it hits
It hits ground with 45° angle when the vertical velocity, vy, is the same as the horizontal velocity, vx, which equals u (initial velocity) if the projectile is fired horizontally and ignoring air-resistance.
vx=u
Let H=given height of cliff
vy²-0²=2gH
vy²=2gH
since vy=u,
u²=2gH
u=√(2gH)
To find the necessary speed at which the projectile must be fired so that it makes a 45-degree angle with the ground when it hits, we can use the principles of projectile motion.
Let's break down the problem step-by-step:
1. Understand the given information:
- The projectile is fired horizontally, which means its initial vertical velocity component is zero.
- The angle at which the projectile hits the ground is 45 degrees.
2. Determine the known quantities:
- The initial vertical velocity component (Vy) is zero.
- The angle at which the projectile hits the ground (θ) is 45 degrees.
- We need to find the initial horizontal velocity component (Vx) or the magnitude of the initial velocity (V).
- Note that the height of the cliff is not necessary to find the initial velocity magnitude.
3. Apply the principles of projectile motion:
In vertical motion:
- The initial vertical velocity component (Vy) is zero.
- The time taken (t) for the projectile to hit the ground is the same as in any other case of free fall.
- The equation for the vertical displacement (h) in terms of time is: h = (1/2) * g * t^2, where g is the acceleration due to gravity (9.8 m/s^2).
In horizontal motion:
- The initial horizontal velocity component (Vx) remains constant throughout the motion because there is no acceleration in the horizontal direction.
- The horizontal displacement (range) of the projectile (R) can be calculated using the equation: R = Vx * t, where t is the time taken to hit the ground.
In our case, we know that the angle at which the projectile hits the ground is 45 degrees. Therefore, we can say that:
- Vx = V * cos(45), where V is the magnitude of the initial velocity (which is what we need to find).
Additionally, we can manipulate the equations to find t:
- From the vertical motion equation, we can rearrange it as: t = √(2h/g).
- Substituting this value of t in the equation for R, we get: R = Vx * √(2h/g).
4. Use the given information to find V:
Since the height of the cliff is not given, we don't have a specific value for h. However, we can select any arbitrary height for the cliff to find the value of V.
For example, let's assume the height of the cliff is 100 meters:
- Plug in the values into the equation for R: R = Vx * √(2 * 100 / 9.8).
- Since the projectile is fired horizontally, the range is equal to the horizontal distance covered.
Thus, the equation can be written as: R = V * cos(45) * √(2 * 100 / 9.8).
- Simplify the equation: R = V * √(2 * 100 / 9.8).
Finally, solve the equation for V by rearranging it:
- V = R / √(2 * 100 / 9.8).
- Substitute the known value for R (horizontal distance) into the equation.
- Calculate V using the equation to get the required speed at which the projectile must be fired to make a 45-degree angle with the ground when it hits.
By following these steps, you can find the required speed at which the projectile must be fired to make a 45-degree angle with the ground when it hits.