A person riding a bicycle on level ground at a speed of 10.6 m/s throws a baseball forward at a speed of 19.1 m/s relative to the bicycle at an angle of 41° relative to the horizontal (x) direction.
(a) If the ball is released from a height of 1.3 m, how far does the ball travel horizontally, as measured from the spot it is released?
(b) How far apart are the bicycle and the ball when the ball lands? Ignore air drag.
To solve this problem, we can break it down into vertical and horizontal components.
(a) First, let's find the time it takes for the ball to land.
The initial vertical velocity (Vy) of the ball can be found using the given angle:
Vy = 19.1 m/s * sin(41°)
Vy = 19.1 m/s * 0.6561
Vy ≈ 12.53 m/s
Next, we can use the displacement equation to find the time of flight (T) for the ball:
Δy = Vy * T + (1/2) * g * T^2
Where Δy is the initial height of the ball (1.3 m) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we have:
1.3 m = 12.53 m/s * T - (4.9 m/s^2) * T^2
Rearranging the equation:
4.9T^2 - 12.53T + 1.3 = 0
Solving this quadratic equation, we find two solutions: T ≈ 0.41 s and T ≈ 2.16 s. Since we are considering the time it takes for the ball to land, we discard the smaller value of T.
Thus, the time it takes for the ball to land is approximately 2.16 seconds.
To calculate the horizontal distance traveled by the ball, we use the formula:
Δx = Vx * T
Where Vx is the initial horizontal velocity of the ball.
The horizontal velocity of the ball (Vx) can be found using the given angle:
Vx = 19.1 m/s * cos(41°)
Vx = 19.1 m/s * 0.7547
Vx ≈ 14.42 m/s
Plugging in the values, we have:
Δx = 14.42 m/s * 2.16 s
Δx ≈ 31.16 m
Therefore, the ball travels approximately 31.16 meters horizontally from the spot it was released.
(b) To find the horizontal distance between the bicycle and the ball when the ball lands, we multiply the velocity of the bicycle (10.6 m/s) by the time it takes for the ball to land (2.16 s):
Distance = Velocity * Time
Distance = 10.6 m/s * 2.16 s
Distance ≈ 22.90 m
So, the bicycle and the ball are approximately 22.90 meters apart when the ball lands.
To find the horizontal distance that the baseball travels, you need to determine the time it takes for the ball to land. Once you have the time, you can calculate the horizontal distance using the formula:
Horizontal distance = (horizontal velocity of the ball) * (time of flight)
Let's break down the problem step by step:
Step 1: Split the initial velocity of the ball into its horizontal and vertical components.
The horizontal component of the initial velocity is given by:
horizontal velocity = velocity * cos(angle)
Substituting the given values:
horizontal velocity = 19.1 m/s * cos(41°)
Step 2: Calculate the time of flight.
The time it takes for the ball to reach the ground can be found using the vertical motion equation:
vertical displacement = (initial vertical velocity) * time - (1/2) * acceleration * time^2
Since the ball is being released from a height of 1.3 m with an initial vertical velocity of 0 m/s, and the acceleration due to gravity is -9.8 m/s^2 (assuming downward as negative):
1.3 m = 0 m/s * time - (1/2) * (-9.8 m/s^2) * time^2
Now, solve the quadratic equation to find the time of flight.
Step 3: Calculate the horizontal distance.
Now that you have the horizontal velocity and the time of flight, you can find the horizontal distance traveled by the ball:
Horizontal distance = horizontal velocity * time of flight
Plug in the values and calculate the distance.
For part (b), since the ball is being thrown horizontally, the horizontal velocity of the ball is the same as the horizontal velocity of the bicycle (10.6 m/s). Thus, the only thing left to find is the time it takes for the ball to land.
Using the same vertical motion equation:
0 m = 0 m/s * time - (1/2) * (-9.8 m/s^2) * time^2
Solve for time to find how long it takes for the ball to land.
Once you have the time, you can calculate the horizontal distance traveled by the bicycle during that time by using the formula:
Horizontal distance = (horizontal velocity of the bicycle) * (time of flight)
Plug in the values and calculate the distance.