Take g= 10 MS^-2. A stone propelled from a catapult with a speed of 50 MS^-1 attains a height of 100 m. Calculate:
a. the time of flight.
b. the angle of projection.
c. the range attained.
To calculate the time of flight, angle of projection, and range attained, we can use the equations of motion for projectile motion.
a. To find the time of flight, we need to consider the vertical motion of the stone. The stone is projected vertically upward and then it falls back to the ground. At the highest point of its trajectory, the vertical velocity becomes zero.
We can use the equation:
v = u + gt
Where:
v = final velocity (0 m/s at the highest point)
u = initial velocity (50 m/s)
g = acceleration due to gravity (10 m/s^2)
t = time taken
Rearranging the equation for time, we have:
t = (v - u) / g
Substituting the values into the equation:
t = (0 - 50) / 10
t = -5 seconds
However, time cannot be negative in this context since we are looking for the time of flight. Therefore, we take the magnitude of the time, which gives:
t = 5 seconds
So, the time of flight is 5 seconds.
b. To find the angle of projection, we can use the horizontal and vertical components of the initial velocity. The stone follows a parabolic trajectory, so the angle of projection is the angle formed between the initial velocity vector and the horizontal direction.
Using basic trigonometry, we have:
tan(theta) = vertical component of velocity / horizontal component of velocity
The horizontal component of velocity (uh) remains constant throughout the motion and is given by:
uh = u * cos(theta)
The vertical component of velocity (uv) changes due to the acceleration of gravity and can be found using:
uv = u * sin(theta) - g * t
Given that uv = 0 at the highest point of the trajectory, we can substitute t = 5 s:
0 = 50 * sin(theta) - 10 * 5
Simplifying the equation, we get:
sin(theta) = 0.5
To find the angle of projection, we take the inverse sine (arcsin) of both sides of the equation:
theta = arcsin(0.5) ≈ 30 degrees
Therefore, the angle of projection is approximately 30 degrees.
c. To find the range attained, we can use the horizontal component of the initial velocity and the time of flight. The range is the horizontal distance covered by the stone.
The range (R) can be calculated using the equation:
R = uh * t
Substituting the values into the equation:
R = (50 * cos(30)) * 5
Using the cosine value (cos(30) ≈ 0.866), we have:
R = 50 * 0.866 * 5
R ≈ 216.5 meters
Therefore, the range attained is approximately 216.5 meters.
To solve this problem, we can use the equations of projectile motion. Let's use the following variables:
v = initial velocity of the stone (50 m/s)
g = acceleration due to gravity (-10 m/s^2, since it's acting against the motion of the stone)
h = maximum height of the stone (100 m)
a. To find the time of flight, we can use the equation for vertical motion:
h = (v^2 * sin^2θ) / (2g)
Rearranging the equation to solve for time, t:
2gh = v^2 * sin^2θ
t = 2v * sinθ / g
t = 2 * 50 * sinθ / (-10)
t = -10 * sinθ
We know that t > 0, so we take the positive value of sinθ:
t ≈ 10 seconds
b. To find the angle of projection, we can use the equation for vertical motion:
v = v₀ * sinθ - g * t
Since the stone has no vertical velocity at the peak of its trajectory, we know that sinθ = 0 at that point. Thus:
0 = v₀ * sinθ - g * t
v₀ * sinθ = g * t
Substituting the known values:
50 * sinθ = 10 * 10
sinθ = 1
θ = arcsin(1)
θ ≈ 90 degrees
Therefore, the angle of projection is approximately 90 degrees.
c. To find the range attained, we can use the equation for horizontal motion:
R = v₀ * cosθ * t
Substituting the known values:
R = 50 * cos90 * 10
R = 0
Therefore, the range attained is 0 meters.
To solve this problem, we can use the equations of motion for projectile motion. Let's break down each part of the problem step-by-step:
a. Finding the time of flight:
The time of flight is the total time it takes for the stone to reach the maximum height and then return back down to the ground. We can find it using the formula:
Time of flight = 2 * (Vertical component of initial velocity) / (acceleration due to gravity)
The vertical component of the initial velocity can be calculated as:
Vertical component of initial velocity = Initial velocity * sin(angle of projection)
Given:
Initial velocity (u) = 50 m/s
Acceleration due to gravity (g) = 10 m/s^2
Let's substitute the given values into the equations:
Vertical component of initial velocity = 50 m/s * sin(angle of projection)
Since we don't know the angle of projection yet, we can't calculate the time of flight at this point.
b. Finding the angle of projection:
The angle of projection (θ) can be determined using the equation:
Vertical component of initial velocity = Initial velocity * sin(angle of projection)
Given:
Vertical component of initial velocity = 50 m/s * sin(angle of projection)
To find the angle, we need to rearrange the equation:
sin(angle of projection) = Vertical component of initial velocity / Initial velocity
Now, substitute the given values into the equation:
sin(angle of projection) = (50 m/s) / 50 m/s
We can simplify this to:
sin(angle of projection) = 1
To find the angle, we take the inverse sine of both sides:
angle of projection = sin^(-1)(1)
The inverse sine of 1 is 90 degrees, so:
angle of projection = 90 degrees
c. Finding the range attained:
The range is the horizontal distance covered by the stone during its flight. We can use the formula:
Range = Horizontal component of initial velocity * Time of flight
The horizontal component of the initial velocity can be calculated as:
Horizontal component of initial velocity = Initial velocity * cos(angle of projection)
Given:
Initial velocity (u) = 50 m/s
Angle of projection (θ) = 90 degrees
Let's substitute the given values into the equation:
Horizontal component of initial velocity = 50 m/s * cos(90 degrees)
cos(90 degrees) = 0, so:
Horizontal component of initial velocity = 50 m/s * 0 = 0 m/s
Since the horizontal component of the initial velocity is 0, the range is also 0.
In summary:
a. The time of flight cannot be calculated without knowing the angle of projection.
b. The angle of projection is 90 degrees.
c. The range attained is 0 meters.