a shot is released at a height of 1.89m, with a speed of 13m/s and at an angle of 34 degrees. Calculate the maximum height reached and the time at which this occurs, the range and the time of flight, and the optimum projection angle.
To calculate the maximum height reached by the shot, we need to find the vertical component of the initial velocity. It can be calculated using the formula:
Vy = V * sin(θ)
Where Vy is the vertical component of the initial velocity, V is the initial speed of the shot, and θ is the angle at which it was released.
Substituting the given values:
Vy = 13 m/s * sin(34°)
Vy ≈ 7.1 m/s
Now, we can calculate the maximum height using the kinematic equation for vertical displacement:
Δy = (Vy^2) / (2 * g)
Where Δy is the displacement in the vertical direction, Vy is the vertical component of the initial velocity, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the values:
Δy = (7.1 m/s)^2 / (2 * 9.8 m/s^2)
Δy ≈ 2.426 m
Therefore, the maximum height reached by the shot is approximately 2.426 meters.
To find the time at which the maximum height occurs, we can use the equation:
t = Vy / g
Substituting the values:
t = 7.1 m/s / 9.8 m/s^2
t ≈ 0.724 s
Hence, the time at which the maximum height occurs is approximately 0.724 seconds.
To calculate the range of the shot, we can use the formula:
R = V * cos(θ) * t
Where R is the range, V is the initial speed, θ is the angle at which it was released, and t is the time of flight.
Substituting the values:
R = 13 m/s * cos(34°) * t
R ≈ 9.757 m
Therefore, the range of the shot is approximately 9.757 meters.
The time of flight can be calculated by doubling the time it takes for the shot to reach its maximum height:
Time of flight = 2 * t ≈ 2 * 0.724 s ≈ 1.448 s
So, the time of flight is approximately 1.448 seconds.
Finally, the optimum projection angle can be determined using the formula:
Optimum projection angle = atan(Vy / Vx)
Where atan is the arctangent function, Vy is the vertical component of the initial velocity, and Vx is the horizontal component of the initial velocity.
Substituting the values:
Optimum projection angle = atan(7.1 m/s / (13 m/s * cos(34°)))
Optimum projection angle ≈ 44.94°
Therefore, the optimum projection angle for achieving the maximum range for this shot is approximately 44.94 degrees.