HOW DO I DO THIS?
A daredevil is shot out of a cannon at 45.0° to the horizontal with an initial speed of 28.0 m/s. A net is positioned a horizontal distance of 51.0 m from the cannon. At what height above the cannon should the net be placed in order to catch the daredevil?
Calculate the time to reach the 51m mark:
t = 51/(28 cos(&theta))
Calculate the elevation of the daredevil
S = ut+(1/2)gt²
= 28sin(θ)+(1/2)(-9.8)t²
Solve for S.
Double-check your results or else the poor (dare)devil will land on hard ground.
To find the height above the cannon where the net should be placed to catch the daredevil, we can use the kinematic equations of projectile motion. Here are the steps to solve this problem:
Step 1: Break down the initial velocity into its horizontal and vertical components.
The initial velocity (v) of the daredevil has two components:
- The horizontal component (v_x) remains constant throughout the motion.
- The vertical component (v_y) changes due to the acceleration due to gravity.
Given:
Initial speed (v) = 28.0 m/s
Launch angle (θ) = 45.0°
To find the horizontal component, we can use the cosine of the launch angle:
v_x = v * cos(θ)
To find the vertical component, we can use the sine of the launch angle:
v_y = v * sin(θ)
Step 2: Determine the time of flight (t).
The time of flight is the time it takes for the daredevil to reach the net horizontally.
We can use the formula for the time of flight in projectile motion:
t = (2 * v_y) / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 3: Calculate the horizontal distance traveled (d).
The horizontal distance traveled by the daredevil can be calculated using the formula:
d = v_x * t
Step 4: Find the height above the cannon.
Since the net is positioned at a horizontal distance (d) of 51.0 m from the cannon, we need to find the corresponding height (h).
We can use the formula for vertical displacement:
h = v_y * t + (1/2) * (-g) * t²
Substitute the values of v_y, t, and g into the formula to find the height (h).
Step 5: Calculate the height.
Substitute the values into the formulas:
v_x = v * cos(θ)
v_y = v * sin(θ)
t = (2 * v_y) / g
d = v_x * t
h = v_y * t + (1/2) * (-g) * t²
v = 28.0 m/s
θ = 45.0°
g = 9.8 m/s²
Calculate v_x, v_y, t, d, and h using the above formulas and the given values to find the height above the cannon where the net should be placed to catch the daredevil.
To answer this question, we can use the principles of projectile motion. The daredevil's motion can be broken down into horizontal and vertical components.
First, let's analyze the horizontal motion. The horizontal component of the daredevil's velocity remains constant throughout the motion. We can use the formula:
horizontal distance = horizontal velocity × time
Since the daredevil is shot out of the cannon horizontally, the horizontal component of the velocity is given by:
horizontal velocity = initial velocity × cos(angle)
Given that the initial velocity is 28.0 m/s and the launch angle is 45.0°, we can calculate the horizontal velocity:
horizontal velocity = 28.0 m/s × cos(45.0°)
Next, let's calculate the time of flight. The total time the daredevil is in the air can be found using the vertical motion. We can use the formula:
time of flight = (2 × vertical velocity) / g
The vertical component of the velocity can be found using:
vertical velocity = initial velocity × sin(angle)
Given the initial velocity and the launch angle, we can calculate the vertical velocity:
vertical velocity = 28.0 m/s × sin(45.0°)
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Now, we can calculate the time of flight:
time of flight = (2 × vertical velocity) / g
Next, let's calculate the maximum height reached by the daredevil. The maximum height is achieved when the vertical velocity reaches zero. We can use the formula:
maximum height = (vertical velocity²) / (2 × g)
Once we have the time of flight and the maximum height reached, we can calculate the vertical distance covered by the daredevil:
vertical distance = maximum height + (vertical velocity × time of flight)
Finally, we can use the Pythagorean theorem to calculate the height above the cannon where the net should be placed. The vertical distance represents the vertical displacement from the cannon to the maximum height, and the horizontal distance is the displacement from the maximum height to the net.
height above the cannon = vertical distance + horizontal distance
By following these steps and substituting the values into the formulas, you should be able to find the height above the cannon where the net should be placed to catch the daredevil.