A ball is thrown at 19.5m/s at 55∘ above the horizontal. Someone located 30m away along the line of the path starts to run just as the ball is thrown. How fast, and in which direction, must the catcher run to catch the ball at the level from which it was thrown?
To determine the speed and direction at which the catcher must run to catch the ball at the level from which it was thrown, we need to analyze the motion of the ball and the relative motion between the catcher and the ball.
Let's break down the problem into different components:
1. Horizontal Motion of the Ball:
Since no external horizontal forces act on the ball, it will maintain a constant velocity horizontally. The initial velocity of the ball is given as 19.5 m/s at an angle of 55 degrees above the horizontal.
We can split the initial velocity into its horizontal and vertical components:
- Horizontal component: 19.5 m/s * cos(55°)
- Vertical component: 19.5 m/s * sin(55°)
2. Vertical Motion of the Ball:
The ball experiences free fall vertically due to the force of gravity. The vertical motion is independent of the horizontal motion. The initial vertical velocity is given as 19.5 m/s * sin(55°), and the acceleration due to gravity is approximately 9.81 m/s² (assuming no air resistance).
3. Relative Motion between the Catcher and the Ball:
As the ball is thrown, the catcher starts running from a distance of 30 m along the line of the ball's path. The objective is to determine the speed and direction at which the catcher must run to meet the ball at the same level.
Let's assume the catcher's speed is v and the angle at which they run is θ. We need to find the values of v and θ.
Considering the time it takes for the ball to travel the horizontal distance of 30 m:
Time = Distance / Horizontal Component of Velocity
= 30 m / (19.5 m/s * cos(55°))
Now, using the time, we can calculate the vertical distance the ball would have fallen during this time:
Vertical Distance = Vertical Component of Velocity * Time + (1/2) * Acceleration * Time^2
= (19.5 m/s * sin(55°)) * Time + (1/2) * 9.81 m/s² * Time^2
Since the catcher needs to meet the ball at the same level, the vertical distance fallen by the ball should equal the vertical displacement of the catcher during this time, which is zero.
Equating the above equation to zero, we get:
(19.5 m/s * sin(55°)) * Time + (1/2) * 9.81 m/s² * Time^2 = 0
By solving this quadratic equation, we can determine the time taken for the ball to reach the catcher's position. Once we have the time, we can calculate the speed and direction at which the catcher must run to meet the ball.
Speed = Distance / Time
Direction = Angle of the Catcher's Path (θ)
By substituting the values into the above formulas, we can calculate the exact values for the speed and direction at which the catcher must run to catch the ball at the level from which it was thrown.