You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 36.761s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?
To find the maximum horizontal range of the dart gun, we can use the equations of motion.
First, let's break down the motion of the dart gun into two parts: the upward trajectory and the downward trajectory.
During the upward trajectory, the initial velocity of the dart gun is zero when it is just shot, and it reaches its highest point when it momentarily stops before coming back down. The time it takes to reach the highest point is half of the total time, which is t/2 = 36.761s / 2 = 18.3805s.
Using the equation for vertical displacement during upward motion:
Δy = vi * t - 1/2 * g * t^2
Where:
Δy: Vertical displacement (in this case, the maximum height reached)
vi: Initial vertical velocity (which is zero, as mentioned before)
t: Time in seconds
g: Acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values:
Δy = 0 * 18.3805 - 1/2 * 9.8 * (18.3805)^2
= -1703.203 m
Now, during the downward trajectory, the only force acting on the dart is gravity, which causes it to accelerate downward at a constant rate. The horizontal component of velocity, however, remains constant throughout.
The horizontal range is the product of the horizontal velocity and the total time of flight.
The horizontal velocity is given by:
vx = distance / time
We need to find the distance traveled in the vertical direction to calculate the horizontal range. The distance traveled vertically is equal to twice the vertical displacement during the upward trajectory since the total displacement is the sum of the upward and downward displacements.
So, the distance traveled vertically is:
dy = 2 * |Δy|
= 2 * |-1703.203|
= 3406.406 m
Now, the horizontal range can be calculated using the equation:
Range = vx * t
Where "vx" is the horizontal velocity and "t" is the total time of flight, which is 36.761s.
Substituting in the values:
Range = (dx / t) * t
= dx
Therefore, the maximum horizontal range of your gun is 3406.406 meters.
To find the maximum horizontal range, we need to know the time it takes for the dart to reach its maximum height. Since the dart is shot straight up and takes 36.761 seconds to return to the barrel, we can assume it takes half of that time to reach its peak. So the time to reach maximum height would be 36.761 / 2 = 18.3805 seconds.
Now, let's use the equations of motion to find the maximum height reached by the dart during this time. We'll use the kinematic equation:
h = (V_0^2 * sin^2θ) / (2 * g),
where h is the maximum height, V_0 is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity.
Since the dart was shot straight up, the launch angle θ is 90 degrees and the sine of 90 degrees is 1. So, the equation simplifies to:
h = V_0^2 / (2 * g).
Now, let's find the initial velocity V_0. We'll use the formula:
V = g * t,
where V is the vertical velocity of the dart and t is the time it takes to reach maximum height. Substituting t = 18.3805 seconds, we can solve for V:
V = g * t = 9.8 m/s^2 * 18.3805 s = 179.9919 m/s.
Now, we can substitute this value back into the equation for the maximum height:
h = V_0^2 / (2 * g) = (179.9919 m/s)^2 / (2 * 9.8 m/s^2) = 1644.9494 m.
The maximum height reached by the dart is approximately 1644.9494 meters. However, we need to find the horizontal range when the dart reaches the ground.
To do this, we can use the equation:
R = V_0 * t,
where R is the horizontal range and t is the total time of flight. In this case, t is twice the time taken to reach maximum height, which is 2 * 18.3805 seconds = 36.761 seconds.
Substituting the values, we get:
R = V_0 * t = 179.9919 m/s * 36.761 s = 6608.5299 m.
So, the maximum horizontal range of your gun is approximately 6608.5299 meters.
if t is RISE time
and Vi is initial speed up
v = Vi - 9.81 t
at top v = 0
so
Vi = 9.81 * 36.761/2
(because it spends half the time going up)
Now I hope you know 45 degrees gives max range
New Vi = Vi sin 45
U = Vi cos 45
with new Vi find time to top same way 9.81 t = Vi
time in air = 2 t
d = u * 2 t