You shoot your potato gun horizontally from shoulder height, which is 190 cm. The potato lands 15 m away from you. You would like the potato to go 90 m, so you climb up a ladder and fire the gun horizontally from there. How high over the ground must the gun be?

To find out how high over the ground the gun must be, we need to calculate the initial height at which the potato was fired.

We will use the projectile motion equations to solve this problem. The horizontal motion of the potato is not affected by the height, so we can ignore it for now. Therefore, only vertical motion needs to be considered.

We know the following:

Initial vertical velocity (vi_y) = 0 m/s (since it is horizontally fired)
Final vertical velocity (vf_y) = 0 m/s (since the potato will fall back to the ground)
Vertical displacement (d_y) = ? (we need to find this)
Acceleration due to gravity (g) = 9.8 m/s^2

The vertical motion equation we will use is: vf_y^2 = vi_y^2 + 2 * g * d_y

Since vf_y is 0, the equation becomes: 0^2 = 0^2 + 2 * 9.8 * d_y

Simplifying, we get: 0 = 19.6 * d_y

Solving for d_y, we find that the vertical displacement is 0 meters. This means that the height over the ground does not affect the horizontal distance traveled by the potato.

Therefore, the height over the ground must be 0 meters.

To solve this problem, we can use the principle of projectile motion. When the potato is shot horizontally, its initial vertical velocity component is zero. The only force acting on the potato is gravity, which will cause it to fall vertically downwards.

Given:
Initial height (shoulder height) = 190 cm = 1.9 m
Horizontal distance (range) = 15 m
Desired range = 90 m

Let's start by finding the initial velocity (V0) of the potato when fired horizontally. The horizontal velocity (Vx) remains constant throughout the motion, while the vertical velocity (Vy) changes due to the force of gravity.

We can calculate the initial velocity (V0) using the formula for the range of projectile motion:

Range (R) = V0 * t,
where t is the time of flight.

Since the potato was shot horizontally, the time of flight can be determined using the vertical component of motion:

Vertical displacement (Y) = 1/2 * g * t^2,
where g is the acceleration due to gravity (9.8 m/s^2).

For the potato to land at a range of 15 m, the vertical displacement is zero. Therefore, we can set the equation for vertical displacement equal to zero:

0 = 1/2 * g * t^2.

Solving for t:
0 = 4.9 * t^2,
t^2 = 0,
t = 0.

Since t = 0, it means that the potato falls vertically at the same time it was shot. This implies that the projectile motion has zero time of flight, and thus it only has vertical motion, making its horizontal velocity zero.

Now, let's calculate the initial velocity (V0) using the formula for the horizontal distance:

Range (R) = V0 * t.
Since t = 0, the equation simplifies to:
Range (R) = V0 * 0,
15 = V0 * 0,
15 = 0.
This equation is not valid because it results in an inconsistency.

From the above calculation, we can conclude that the potato will not reach the desired range of 90 m when shot horizontally from shoulder height.

To calculate the required height (h) from which the potato should be shot to achieve a range of 90 m, we need to determine the initial vertical velocity (Vy) required.

Let's assume the time of flight for the 90 m range is t.

Using the equation for vertical displacement:
Vertical displacement (Y) = 1/2 * g * t^2.

Since the initial vertical velocity is zero (since the potato shoots horizontally), the equation simplifies to:
Y = 1/2 * g * t^2.

The vertical displacement can be calculated by taking the difference in heights:
Y = h - shoulder height.

Therefore:
h - shoulder height = 1/2 * g * t^2.

We can now solve for h:
h = 1/2 * g * t^2 + shoulder height.

Let's calculate the value:
h = 1/2 * 9.8 * t^2 + 1.9.

To find the value of t, we can use the formula for the range of projectile motion:
Range (R) = Vx * t,
R = Vx * t,
90 = Vx * t.

Since the horizontal velocity (Vx) remains constant throughout the motion, we can express it as:
Vx = V0.

We don't yet know the value of V0, but we can solve for it using the range equation and the desired range:
90 = V0 * t.

We established earlier that the horizontal velocity remains constant while the vertical velocity is zero. Therefore, V0 = 90 / t.

Substituting this expression for V0 in the equation for height:
h = 1/2 * 9.8 * t^2 + 1.9.

Finally, substitute this value of h into the equation:
90 / t = 1/2 * 9.8 * t^2 + 1.9.

Now, solve this equation to find the value of t, and then calculate the value of h using the equation h = 1/2 * 9.8 * t^2 + 1.9.