A mortar elevated to 60 degrees scores a hit 15 m up a building 30m away. with what speed was the shell fired?
You have two equations, vertical and horizontal..
horizontal
30=Vcos60 * time solve for time in terms of V.
Verical.
15=Vsin60*time-4.9 time^2
put the expression for time you got above, and then solve for V. It is a quadratic, and there is some messy algebra.
To calculate the initial velocity at which the mortar shell was fired, we can use the equations of motion and trigonometry.
Step 1: Determine the vertical and horizontal components of the velocity.
The vertical component of the velocity can be calculated using the equation of motion:
Vertical component of velocity = V * sin(θ),
where V is the initial velocity and θ is the launch angle (60 degrees in this case).
The horizontal component of the velocity can be calculated using the equation of motion:
Horizontal component of velocity = V * cos(θ).
Step 2: Determine the time of flight for the shell to reach the building.
The time of flight can be calculated using the horizontal distance traveled and the horizontal component of velocity:
Time of flight = Distance / Horizontal component of velocity.
Step 3: Determine the total time taken for the shell to reach the building.
Since the vertical component of velocity is affected by the acceleration due to gravity, we can use the equation of motion:
Vertical distance traveled = Vertical component of velocity * Time of flight - (1/2) * g * (Time of flight)^2,
where g is the acceleration due to gravity (9.8 m/s^2).
Substituting the given values into the equation, we can find the total time taken.
Step 4: Calculate the vertical component of velocity using the vertical distance traveled.
The vertical distance traveled is given as 15 m. Rearrange the equation from step 3 to get:
Vertical component of velocity = (Vertical distance traveled + (1/2) * g * (Time of flight)^2) / Time of flight.
Step 5: Calculate the initial velocity.
The initial velocity can be found using the Pythagorean theorem:
V = √((Vertical component of velocity)^2 + (Horizontal component of velocity)^2).
Now, we can plug in the given values into these equations and calculate the initial velocity.
Vertical component of velocity = V * sin(60°) = V * 0.866.
Horizontal component of velocity = V * cos(60°) = V * 0.5.
Time of flight = 30 m / (V * 0.5).
Vertical distance traveled = 15 m.
g = 9.8 m/s^2.
Using the equation from step 4:
(Vertical component of velocity) = (15 m + (1/2) * 9.8 m/s^2 * (Time of flight)^2) / Time of flight.
By substituting the values into the equation, we can find the Vertical component of velocity.
Finally, we can substitute the Vertical and Horizontal components of velocity into the equation from step 5:
Initial velocity = √((Vertical component of velocity)^2 + (Horizontal component of velocity)^2).
Using these calculations, we can determine the speed at which the shell was fired.
To find the speed at which the shell was fired, we can use the principles of projectile motion. Projectile motion occurs when an object is launched into the air and moves along a curved path under the influence of gravity. In this case, the mortar shell is launched at an angle of 60 degrees and lands 15 m vertically above its launch point and 30 m horizontally away.
We can use the following equations of projectile motion to solve for the initial speed (v) of the shell:
1. Vertical motion equation:
Δy = v * sin θ * t - (1/2) * g * t^2
where:
- Δy is the vertical displacement (15 m),
- v is the initial speed,
- θ is the launch angle (60 degrees),
- t is the time of flight, and
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Horizontal motion equation:
Δx = v * cos θ * t
where:
- Δx is the horizontal displacement (30 m).
We can find the value of t by rearranging the second equation:
t = Δx / (v * cos θ)
Substituting the value of t back into the first equation, we get:
Δy = (v * sin θ) * (Δx / (v * cos θ)) - (1/2) * g * (Δx / (v * cos θ))^2
Simplifying the equation further, we have:
Δy = (v * sin θ * Δx) / (v * cos θ) - (1/2) * g * (Δx / v * cos θ)^2
Now, we can substitute the known values into the equation:
15 = (v * sin 60° * 30) / (v * cos 60°) - (1/2) * 9.8 * (30 / v * cos 60°)^2
Simplifying this equation and solving for v, we can find the speed at which the shell was fired.
Note: Make sure to use radians when inputting angles into trigonometric functions in a calculator. To convert degrees to radians, multiply by π/180. For example, sin 60° would be sin(60 * π/180) in your calculator.