A ball is projected horizontally from the edge
of a table that is 1.2 m high, and it strikes the
floor at a point 1.11 m from the base of the
table.The acceleration of gravity is 9.8 m/s2 .How high is the ball above the floor when its
velocity vector makes a −54.3153� angle with
the horizontal?
Answer in units of m
To find the height of the ball above the floor when its velocity vector makes a -54.3153 degree angle with the horizontal, we can use the following steps:
Step 1: Find the initial velocity (v0) of the ball vertically.
Since the ball is projected horizontally, its initial vertical velocity (v0y) is 0 m/s. This is because there is no initial vertical motion.
Step 2: Find the initial velocity (v0) of the ball horizontally.
The horizontal distance traveled by the ball is 1.11 m in the given time of flight. Let's denote the time of flight as "t".
Using the equation: distance = velocity x time
1.11 m = v0x * t
Since the ball is projected horizontally, the initial horizontal velocity (v0x) remains constant throughout the motion. Therefore, v0x = 1.11 m / t.
Step 3: Find the initial velocity (v0) of the ball.
The initial velocity (v0) can be found using the Pythagorean theorem, as follows:
v0 = √(v0x^2 + v0y^2)
= √(1.11 m / t)^2 + 0^2
= √(1.11^2 / t^2)
= 1.11 / t
Step 4: Find the time of flight (t) of the ball.
To determine the time of flight, we can use the equation for vertical displacement:
y = v0yt + (1/2)gt^2
Since the vertical displacement (y) is the height of the table (1.2 m) and the initial vertical velocity (v0y) is 0 m/s, the equation becomes:
1.2 m = (1/2)gt^2
Solving for t:
t^2 = (2 * 1.2 m) / g
t^2 = 2.4 m / g
t = √(2.4 m / g)
Step 5: Find the initial horizontal velocity (v0x) of the ball.
Using the value of t obtained in step 4, we can find v0x:
v0x = 1.11 m / t
Step 6: Find the height of the ball above the floor.
Using the value of v0 obtained in step 3 and the angle of -54.3153 degrees, we can find the vertical component of the velocity:
v0y = v0 * sin(angle)
The height of the ball above the floor is given by the equation:
height = v0y * t + (1/2)gt^2
Substituting the values we obtained in the previous steps:
height = (v0 * sin(angle)) * t + (1/2)g * t^2
Finally, substituting the values of v0, angle, g, and t, we can calculate the height of the ball above the floor. However, since you didn't provide the angle in the previous text, I can't proceed with the calculation without the angle value.
To find the height of the ball above the floor when its velocity vector makes a specific angle with the horizontal, we can use the following steps:
Step 1: Find the initial horizontal velocity of the ball (Vx).
Since the ball is projected horizontally, its initial vertical velocity (Vy) is 0. Therefore, the initial velocity vector of the ball is entirely horizontal. Vx is the initial horizontal velocity.
Step 2: Find the time it takes for the ball to reach the point where the velocity vector makes the given angle.
The time (t) can be found using the equation d = Vx * t, where d is the horizontal distance traveled by the ball before reaching the specific point.
Step 3: Find the vertical distance (h) traveled by the ball from the table to the point.
The vertical distance (h) can be found using the equation h = 0.5 * g * t^2, where g represents the acceleration due to gravity (9.8 m/s^2).
Step 4: Subtract the vertical distance from the table height to find the height of the ball above the floor.
The height above the floor is given by the equation H = Table height - h.
Let's plug in the given values and calculate the height above the floor:
Table height = 1.2 m
Horizontal distance (d) = 1.11 m
Angle with the horizontal = -54.3153 degrees
Acceleration due to gravity (g) = 9.8 m/s^2
Step 1: The initial horizontal velocity (Vx) is the same as the speed of the ball because it is projected horizontally. So, we can calculate Vx using the formula Vx = d / t.
Vx = 1.11 m / t
Step 2: Find the time (t) by using trigonometry to relate the angle and the horizontal and vertical components of velocity.
We know that tan(angle) = Vy / Vx.
Here, angle = -54.3153 degrees, and Vy = 0 (since the ball is projected horizontally).
So, tan(-54.3153 degrees) = 0 / Vx
Solving for Vx:
Vx = 0, which means Vx = d / t = 0.
Therefore, the time (t) is infinite, indicating that the ball will never reach the specified point.