A physics book slides off a horizontal table top with a speed of 1.30 m/s . It strikes the floor after a time of 0.380s . Ignore air resistance.
Find the height of the table top above the floor.
Find the horizontal distance from the edge of the table to the point where the book strikes the floor
Find the horizontal component of the book's velocity just before the book reaches the floor
Find the vertical component of the book's velocity just before the book reaches the floor.
Find the magnitude of the book's velocity just before the book reaches the floor
Find the direction of the book's velocity just before the book reaches the floor.
Express your answer as an angle measured below the horizontal
vertical problem:
0 = h + 0 -4.9 t^2
0 = h - 4.9(.38)^2
h = .708 meters high
u = 1.30 forever
d = u t = 1.3 * .38 = .494 meters away
HORIZONTAL VELOCITY DOES NOT CHANGE !!!
(there is no horizontal force.)
Hey, you know this. You are making me do things you can do yourself.
708meters and 494meters
To find the height of the table top above the floor, we can use the kinematic equation:
h = (1/2)gt^2
Where:
h = height of the table top
g = acceleration due to gravity (9.8 m/s^2)
t = time taken for the book to reach the floor (0.380 s)
Plugging in the values, we get:
h = (1/2)(9.8 m/s^2)(0.380 s)^2
= 0.68 m
Therefore, the height of the table top above the floor is 0.68 meters.
To find the horizontal distance from the edge of the table to the point where the book strikes the floor, we can use the kinematic equation:
d = v*t
Where:
d = horizontal distance
v = horizontal velocity (which remains constant throughout the motion)
t = time taken for the book to reach the floor (0.380 s)
Since the book slides off the table with a horizontal velocity, the horizontal distance is given by:
d = (1.30 m/s)(0.380 s)
= 0.494 m
Therefore, the horizontal distance from the edge of the table to the point where the book strikes the floor is 0.494 meters.
To find the horizontal component of the book's velocity just before it reaches the floor, we know that the horizontal velocity remains constant. Therefore, it is still 1.30 m/s.
To find the vertical component of the book's velocity just before it reaches the floor, we can use the formula:
v = u + gt
Where:
v = final vertical velocity (which is 0 m/s as the book hits the ground)
u = initial vertical velocity (which can be found using the formula v = u + gt)
g = acceleration due to gravity (9.8 m/s^2)
t = time taken for the book to reach the floor (0.380 s)
Rearranging the formula, we get:
u = v - gt
= 0 - (9.8 m/s^2)(0.380 s)
= -3.724 m/s
Therefore, the vertical component of the book's velocity just before it reaches the floor is -3.724 m/s. The negative sign indicates that the book is moving downwards.
To find the magnitude of the book's velocity just before it reaches the floor, we can use the Pythagorean theorem:
v = sqrt(v_horizontal^2 + v_vertical^2)
Where:
v_horizontal = horizontal velocity (1.30 m/s)
v_vertical = vertical velocity (-3.724 m/s)
Plugging in the values, we get:
v = sqrt((1.30 m/s)^2 + (-3.724 m/s)^2)
= 3.92 m/s
Therefore, the magnitude of the book's velocity just before it reaches the floor is 3.92 m/s.
To find the direction of the book's velocity just before it reaches the floor, we can use the formula:
θ = tan^(-1)(v_vertical / v_horizontal)
Where:
θ = angle measured below the horizontal
v_vertical = vertical velocity (-3.724 m/s)
v_horizontal = horizontal velocity (1.30 m/s)
Plugging in the values, we get:
θ = tan^(-1)(-3.724 m/s / 1.30 m/s)
= -71.5°
Therefore, the direction of the book's velocity just before it reaches the floor is 71.5° below the horizontal.
To solve the given problem, we will use the equations of motion, specifically those related to free fall and projectile motion.
1. Find the height of the table top above the floor:
To find the height of the table top, we can use the formula for the displacement in vertical motion:
h = (1/2) * g * t^2
Where:
h is the height
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time taken to reach the floor (0.380 s)
Substituting the values, we have:
h = (1/2) * 9.8 * (0.380)^2
h = 0.68804 m
Therefore, the height of the table top above the floor is approximately 0.688 m.
2. Find the horizontal distance from the edge of the table to the point where the book strikes the floor:
Since there is no horizontal force acting on the book, it will maintain a constant horizontal velocity. The horizontal distance can be calculated using the equation:
d = v * t
Where:
d is the horizontal distance
v is the horizontal velocity (which remains constant)
t is the time taken to reach the floor (0.380 s)
Since the book slides off the table with an initial horizontal velocity, the horizontal distance is:
d = 1.30 * 0.380
d = 0.494 m
Therefore, the horizontal distance from the edge of the table to the point where the book strikes the floor is approximately 0.494 m.
3. Find the horizontal component of the book's velocity just before reaching the floor:
Since there is no horizontal force acting on the book, the horizontal component of its velocity will remain constant throughout its motion. Therefore, the value will be 1.30 m/s, the initial horizontal velocity.
4. Find the vertical component of the book's velocity just before reaching the floor:
To find the vertical component of the book's velocity just before reaching the floor, we can use the formula:
v = u + g * t
Where:
v is the final vertical velocity
u is the initial vertical velocity (which is 0)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time taken to reach the floor (0.380 s)
Substituting the values, we have:
v = 0 + 9.8 * 0.380
v = 3.724 m/s
Therefore, the vertical component of the book's velocity just before reaching the floor is approximately 3.724 m/s.
5. Find the magnitude of the book's velocity just before reaching the floor:
To find the magnitude of the book's velocity just before reaching the floor, we can use the Pythagorean theorem. Since the horizontal and vertical components of the velocity are perpendicular, we can calculate the magnitude using:
|V| = sqrt(v_h^2 + v_v^2)
Where:
|V| is the magnitude of the velocity
v_h is the horizontal component of the velocity (1.30 m/s)
v_v is the vertical component of the velocity (3.724 m/s)
Substituting the values, we have:
|V| = sqrt(1.30^2 + 3.724^2)
|V| = sqrt(1.69 + 13.853776)
|V| = sqrt(15.543776)
|V| = 3.946 m/s
Therefore, the magnitude of the book's velocity just before reaching the floor is approximately 3.946 m/s.
6. Find the direction of the book's velocity just before reaching the floor:
To find the direction, we can use the tangent inverse of the vertical component divided by the horizontal component:
angle = tan^(-1)(v_v / v_h)
Where:
angle is the angle measured below the horizontal
v_h is the horizontal component of the velocity (1.30 m/s)
v_v is the vertical component of the velocity (3.724 m/s)
Substituting the values, we have:
angle = tan^(-1)(3.724 / 1.30)
angle = tan^(-1)(2.864615)
Using a calculator, we find:
angle ≈ 68.85 degrees
Therefore, the direction of the book's velocity just before reaching the floor is approximately 68.85 degrees measured below the horizontal.