In the absence of air resistance, a projectile is launched from and returns to ground level. It follows a trajectory similar to that shown in Figure 3.10 and has a range of 29 m. Suppose the launch speed is doubled, and the projectile is fired at the same angle above the ground. What is the new range?
To solve this problem, we can use the range equation for projectile motion:
Range = (initial velocity)^2 * sin(2θ) / g
where:
- Range is the horizontal distance traveled by the projectile
- initial velocity is the speed of the projectile at the time of launch
- θ is the launch angle
- g is the acceleration due to gravity
Let's assume the initial velocity is V and the launch angle is θ.
Given that the range is 29 m, we can write:
29 = V^2 * sin(2θ) / g
Now, let's double the initial velocity: V' = 2V
Using the new initial velocity, we can determine the new range, R':
R' = (V')^2 * sin(2θ) / g
= (2V)^2 * sin(2θ) / g
= 4V^2 * sin(2θ) / g
Substituting the value of sin(2θ) from the first equation, we have:
R' = 4V^2 * (29g / V^2) / g
= 116g
Therefore, the new range is 116 times the range of the original projectile, or 116 * 29 = 3364 m.
To find the new range when the launch speed is doubled, we need to understand the relationship between the range and launch speed.
The range of a projectile can be calculated using the equation:
Range = (Launch speed)^2 * sin(2*Launch angle) / Gravitational acceleration
In this case, assuming the launch angle remains the same, we want to find the new range when the launch speed is doubled. Let's call the original launch speed "v" and the new launch speed "2v".
So, we can rewrite the equation for the new range as:
New Range = (2v)^2 * sin(2*Launch angle) / Gravitational acceleration
Now, to find the new range, we need to substitute the values into the equation.
Given that the original range is 29 m, we can substitute the original launch speed "v" into the equation:
29 = v^2 * sin(2*Launch angle) / Gravitational acceleration
Next, we double the launch speed and calculate the new range:
New Range = (2v)^2 * sin(2*Launch angle) / Gravitational acceleration
Substitute the value of "v" into the equation:
New Range = (2 * 29)^2 * sin(2*Launch angle) / Gravitational acceleration
Simplifying this equation will give us the new range.
Please note that we need to know the value of the launch angle and gravitational acceleration to get a specific numerical answer.
In the vertical:
vf=vi-gt where t is time in air.
because vf=-vi in the vertical
t=2vi/g
Now, horizontal distance
d=vih*t=vih*2vi/g
now if vi is doubled, vi (in the vertical and horizontal) are doubled, so
d= is 2^2 as far, or four times as far. it is in the air twice as long, and its horizontal velocity is doubled.