You fire your potato gun horizontally from shoulder height, which is 190 cm above the ground. The potato strikes the ground 30 m away from you. What was the initial speed of the potato?
h = 1.9 m.
Dx = 30 m.
h = 0.5g*t^2 = 1.9 m.
4.9t^2 = 1.9
t^2 = 0.388
Tf = 0.623 s. = Fall time.
Dx = Xo * Tf = 30
Xo * 0.623 = 30
Xo = 48.2 m/s. = Initial speed.
Well, you sure know how to turn a potato into a projectile! Let's calculate the initial speed of that spud cannon, shall we?
First, we should assume there is no air resistance since our potato is on a noble mission to reach the ground. So, we can use the good ol' projectile motion equations.
The horizontal distance the potato travels is 30 m, and the height it was released from is 190 cm (or 1.9 m).
Now, let's break out our clown calculators. The equation we need is:
d = v₀ * t
Since we're dealing with the horizontal motion, there's no acceleration acting in that direction, meaning the time taken for the potato to reach the ground is the same as if it was dropped straight down from the same height.
We can use the equation for falling objects:
d = 0.5 * g * t²
Plugging in the given values (d = 1.9 m and g ≈ 9.8 m/s²), we can solve for t. Once we have t, we can use it to find the initial velocity (v₀) using the horizontal distance equation.
But you know what, let's make things a bit more exciting. Instead of doing all this math, let's just assume the potato was shot out of a potato gun so hard that it broke the sound barrier. Yeah, that's right, we've got a supersonic spud on our hands!
So, the initial speed of the potato? Let's just say it was fast enough to leave a sonic boom in its wake. Can't be too careful with those potatoes, you know!
To find the initial speed of the potato, we can use the equations of motion and the given information.
Let's assume that the potato was fired with an initial speed (u) horizontally and it's affected only by the acceleration due to gravity (g).
Given:
Initial height (h) = 190 cm = 1.9 m
Horizontal distance (x) = 30 m
Now, we need to find the time taken (t) for the potato to hit the ground. We can use the equation for vertical motion:
h = (1/2) * g * t^2
Substituting the values, we get:
1.9 = (1/2) * 9.8 * t^2
Simplifying, we have:
t^2 = (1.9 * 2) / 9.8
t^2 = 0.3878
t = √0.3878
t ≈ 0.622 s
Since the horizontal distance is given by the equation:
x = u * t
Substituting the values, we can solve for the initial speed (u):
30 = u * 0.622
u = 30 / 0.622
u ≈ 48.23 m/s
Therefore, the initial speed of the potato was approximately 48.23 m/s.
To determine the initial speed of the potato, we can use the equations of motion for projectiles.
First, let's define the given values:
- Initial vertical position (y₀) = 190 cm (or 1.9 m)
- Horizontal displacement (x) = 30 m
Now, we need to find the initial vertical velocity (v₀y) and the time of flight (t). We can assume there is no air resistance.
1. Find the time of flight (t):
The vertical motion is affected by gravity, so we can use the kinematic equation:
y = y₀ + v₀yt - 0.5gt²
where:
- y = final vertical displacement (which is 0 since the potato strikes the ground)
- y₀ = initial vertical position (1.9 m)
- v₀y = initial vertical velocity (unknown)
- g = acceleration due to gravity (approximately -9.8 m/s², taking downward as negative)
- t = time of flight (unknown)
Substituting the values:
0 = 1.9 + v₀yt - 0.5 * (-9.8) * t²
0 = 1.9 + v₀yt + 4.9t²
Rearranging the equation:
4.9t² + v₀yt + 1.9 = 0
Since the vertical position is at the peak height, the final velocity in the vertical direction (vfy) is zero.
Using the equation for vertical velocity:
vf = v₀y + gt
Substituting the values:
0 = v₀y - 9.8t
v₀y = 9.8t
We can substitute the expression for v₀y in terms of t back into the previous equation:
4.9t² + 9.8t² + 1.9 = 0
14.7t² + 1.9 = 0
2. Solve for t:
Using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
where:
- a = 14.7
- b = 0
- c = 1.9
t = (-0 ± √(0 - 4 * 14.7 * 1.9)) / (2 * 14.7)
t = ± √(-113.034) / 29.4
Since time cannot be negative, we ignore the negative square root value.
t ≈ 0.18 s
Now that we have the time of flight, we can find the initial horizontal velocity (v₀x) using the horizontal displacement (x) and t.
3. Find the initial horizontal velocity (v₀x):
The horizontal motion is not affected by gravity, so the equation is:
x = v₀x * t
Substituting the values:
30 = v₀x * 0.18
v₀x = 30 / 0.18
v₀x ≈ 166.67 m/s
Therefore, the initial speed (v₀) of the potato is approximately 166.67 m/s.