A shell is fired from the ground with an initial speed of 1.62 ✕ 103 m/s at an initial angle of 58° to the horizontal.
(a) Neglecting air resistance, find the shell's horizontal range.
as you know, the range is
r = (v^2 sin2θ)/g
Now just plug in your numbers
frf
To find the shell's horizontal range, we need to analyze the horizontal and vertical components of its initial velocity.
Given:
Initial speed (magnitude of velocity), v = 1.62 ✕ 10^3 m/s
Initial angle, θ = 58°
First, let's break down the initial velocity into horizontal and vertical components. The horizontal component of the velocity (v_x) represents the velocity in the x-axis, while the vertical component (v_y) represents the velocity in the y-axis.
The horizontal component, v_x, can be found using:
v_x = v * cos(θ)
The vertical component, v_y, can be found using:
v_y = v * sin(θ)
Now, let's calculate these values:
v_x = v * cos(θ)
= (1.62 ✕ 10^3 m/s) * cos(58°)
≈ (1.62 ✕ 10^3 m/s) * 0.532
≈ 861.84 m/s
v_y = v * sin(θ)
= (1.62 ✕ 10^3 m/s) * sin(58°)
≈ (1.62 ✕ 10^3 m/s) * 0.848
≈ 1373.76 m/s
Since air resistance is neglected, the shell's motion in the horizontal direction is not affected. The horizontal range (R) is the distance traveled by the shell in the horizontal direction before hitting the ground.
To find the horizontal range, we can use the formula:
R = (v_x * t)
Since the shell will hit the ground when its height (vertical distance) is zero, we can calculate the time (t) it takes to reach this point using the equation of motion in the vertical direction:
h = v_y * t - (1/2) * g * t^2
Where:
h = initial height (assumed to be zero since fired from the ground)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Setting h = 0 and rearranging the equation, we get:
(1/2) * g * t^2 = v_y * t
Simplifying this equation, we get:
(1/2) * g * t = v_y
Solving for t, we have:
t = (2 * v_y) / g
Now, let's calculate t:
t = (2 * v_y) / g
= (2 * 1373.76 m/s) / 9.8 m/s^2
≈ 278.94 s
Now we can substitute the calculated v_x and t values into the equation for the horizontal range (R):
R = (v_x * t)
= (861.84 m/s) * (278.94 s)
≈ 240,473.9 m
Therefore, neglecting air resistance, the shell's horizontal range is approximately 240,473.9 meters.