The initial kinetic energy imparted to a
0.72 kg bullet is 1777 J. The acceleration of gravity is 9.81m/s2.Neglecting air resistance, find the range of this projectile when it is fired at an angle such that the range equals the maximum height attained.
Answer in units of km.
First calculate the initial velocity Vo, using
(M/2)Vo^2 = 1777 J
If the range equals the maximum height,
2(Vo^2/g)sinA*cosA = (1/2)(Vo^2/g)sin^2A
2 cosA = (1/2)sin A
tanA = 4 A = 76.0 degrees
Use that angle in either the range or the max height formula
To solve this problem, we need to use principles of projectile motion.
Let's break down the problem step-by-step:
Step 1: Find the maximum height attained by the projectile.
- The maximum height occurs when the vertical component of the velocity is zero.
- The initial vertical velocity is given by: V_y0 = V₀ * sin(θ), where V₀ is the initial velocity and θ is the angle of projection.
- At the highest point, V_y = 0, so we have: V_y = V_y0 - g * t = 0, where g is the acceleration due to gravity and t is the time taken to reach maximum height.
- Rearranging the equation, we get: t = V_y0 / g.
- The time taken to reach maximum height is the same as the time taken for the projectile to return to the ground, so the total flight time (T) is twice the time taken to reach maximum height: T = 2t.
Step 2: Find the range of the projectile.
- The horizontal component of the velocity remains constant throughout the motion.
- The horizontal velocity is given by: V_x = V₀ * cos(θ).
- The range (R) of the projectile can be calculated using the equation: R = V_x * T.
Step 3: Convert the range from meters to kilometers.
- Divide the range (R) by 1000 to convert it from meters to kilometers.
Now let's plug in the given values and solve the problem:
Given:
Bullet mass (m) = 0.72 kg
Initial kinetic energy (K) = 1777 J
Acceleration due to gravity (g) = 9.81 m/s^2
Step 1:
Using the formula for kinetic energy: K = (1/2) * m * V₀^2
1777 = (1/2) * 0.72 * V₀^2
V₀^2 = (2 * 1777) / 0.72
V₀ = sqrt((2 * 1777) / 0.72)
Step 2:
Using the formula for maximum height: H = (V₀ * sin(θ))^2 / (2 * g)
Using the formula for range: R = V₀ * cos(θ) * T
Since we want the range to be equal to the maximum height, we can equate the equations:
R = (V₀ * sin(θ))^2 / (2 * g)
(R * 2 * g) = (V₀ * sin(θ))^2
V₀^2 * sin^2(θ) = (R * 2 * g)
sin^2(θ) = ((R * 2 * g) / V₀^2)
θ = arcsin(sqrt((R * 2 * g) / V₀^2))
Step 3:
Convert the range from meters to kilometers: R_km = R / 1000
Now you can substitute the given values into the equations to find the angle and range.
To find the range of the projectile when it is fired at an angle such that the range equals the maximum height attained, we can use the following steps:
Step 1: Find the maximum height attained by the projectile.
Step 2: Determine the time taken to reach the maximum height.
Step 3: Use the time obtained in Step 2 to calculate the total time of flight.
Step 4: Use the total time of flight to calculate the range.
Let's begin with Step 1:
Step 1: Find the maximum height attained by the projectile.
The initial kinetic energy of the bullet can be equated to the potential energy at maximum height.
Kinetic Energy = Potential Energy
Using the given values:
Initial kinetic energy (K) = 1777 J
Mass of the bullet (m) = 0.72 kg
Acceleration due to gravity (g) = 9.81 m/s^2
K = mgh
where h is the maximum height.
Rearranging the equation to solve for h:
h = K / (mg)
Substituting the values:
h = 1777 J / (0.72 kg * 9.81 m/s^2)
h ≈ 250.38 m
Step 1 Result: The maximum height attained by the projectile is approximately 250.38 meters.
Moving on to Step 2:
Step 2: Determine the time taken to reach the maximum height.
The time taken to reach the highest point of the trajectory can be calculated using the formula:
Time of ascent (t) = (Initial vertical velocity) / (Acceleration due to gravity)
Since the projectile is fired at an angle such that the range equals the maximum height attained, the initial vertical velocity is zero.
Therefore, t = 0.
Step 2 Result: The time taken to reach the maximum height is 0 seconds.
Next is Step 3:
Step 3: Calculate the total time of flight.
The total time of flight is the sum of the time taken to reach the maximum height (t) and the time taken to descend from the maximum height (t).
Total time of flight = t + t
Since t = 0 (as calculated in Step 2), the total time of flight is also zero.
Step 3 Result: The total time of flight is 0 seconds.
Finally, Step 4:
Step 4: Calculate the range.
The range of the projectile can be determined using the formula:
Range = Initial horizontal velocity × Total time of flight
Since the projectile is fired at an angle such that the range equals the maximum height attained, the initial horizontal velocity is the only component that contributes to the range.
Given that the initial kinetic energy is equal to the initial kinetic energy in the horizontal direction:
Initial kinetic energy (K) = 1777 J
Kinetic energy (K) = 0.5 × mass of the bullet × (initial horizontal velocity)^2
Solving for initial horizontal velocity:
(initial horizontal velocity) ^ 2 = (2K) / m
initial horizontal velocity = √((2K) / m)
Substituting the values:
initial horizontal velocity = √((2 * 1777 J) / 0.72 kg)
initial horizontal velocity ≈ 53.34 m/s
Range = initial horizontal velocity × total time of flight
Range ≈ 53.34 m/s × 0 s
Range ≈ 0 meters
Step 4 Result: The range of the projectile is approximately 0 meters.
In conclusion, the range of the projectile when it is fired at an angle such that the range equals the maximum height attained is approximately 0 meters.