In her physics lab, Jessica rolls a 11-g marble down a ramp and off the table with a horizontal velocity of 1.38 m/s. The marble falls in a cup placed 0.45 m from the table's edge. Find the table's height.
Time of flight = t
1.38 t = 0.45
t = 0.326 s
Now calculate how far the marble falls in 0.326 s. That is the table's height, H.
H = (g/2)t^2
To find the table's height, we need to break down the problem into steps and use the principles of physics and kinematics.
Step 1: Determine the time taken for the marble to reach the cup.
We can use the horizontal velocity and the distance to find the time taken.
Given:
Horizontal velocity, v = 1.38 m/s
Horizontal distance, d = 0.45 m
We can use the formula: d = v * t
Substituting the given values, we have: 0.45 m = 1.38 m/s * t
Rearranging the equation, we find: t = 0.45 m / 1.38 m/s
Calculating the value, we have: t ≈ 0.326 s
Step 2: Determine the vertical distance traveled by the marble when it hits the cup.
In this case, we know that the vertical distance is due to the vertical motion of the marble falling.
We can use the formula for vertically distributing motion: h = (1/2) * g * t^2
Where:
h = vertical distance
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time taken (0.326 s)
Substituting the values, we get: h ≈ (1/2) * 9.8 m/s^2 * (0.326 s)^2
Calculating the value, we have: h ≈ 0.512 m^2/s^2
Step 3: Determine the table's height.
Now, we have the vertical distance traveled by the marble. However, this distance includes both the table's height and the additional vertical distance from the table to the cup. To find the table's height, we need to subtract the additional vertical distance.
Given:
Distance to the cup from the table's edge, d = 0.45 m
Subtracting the additional vertical distance, we have:
Table's height = h - d
Table's height ≈ 0.512 m^2/s^2 - 0.45 m
Calculating the value, we have: Table's height ≈ 0.062 m
Therefore, the table's height is approximately 0.062 meters.
To find the height of the table, we need to use the principles of projectile motion.
First, let's break down the motion of the marble into horizontal and vertical components.
In the horizontal direction, the marble has a constant horizontal velocity of 1.38 m/s. This means there is no horizontal acceleration, and the horizontal distance traveled, x, can be calculated using the formula:
x = v * t,
where v is the horizontal velocity and t is the time taken.
In the vertical direction, the marble is subject to the acceleration due to gravity. The vertical motion can be described using the formula:
y = h + v0 * t + 0.5 * g * t^2,
where y is the vertical displacement, h is the initial height of the table, v0 is the vertical component of the initial velocity, t is the time taken, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the marble falls into the cup placed 0.45 m from the table's edge, this horizontal distance traveled, x, is equal to 0.45 m.
Now, let's find the time it takes for the marble to travel horizontally from the table to the cup. We can use the formula:
x = v * t,
where x is the horizontal distance traveled, v is the horizontal velocity, and t is the time taken.
Rearranging the equation, we can solve for t:
t = x / v.
Plugging in the values, we have:
t = 0.45 m / 1.38 m/s = 0.326 s.
Now, using the time calculated, we can find the vertical displacement, y. Since the marble falls vertically under the influence of gravity, the vertical component of the initial velocity, v0, is 0.
y = h + 0.5 * g * t^2.
Plugging in the values, we have:
0.45 m = h + 0.5 * 9.8 m/s^2 * (0.326 s)^2.
Solving for h, we have:
h = 0.45 m - 0.5 * 9.8 m/s^2 * (0.326 s)^2.
Calculating this expression will give you the height of the table.