The coach throws a baseball to a player with an initial speed of `20 m//s` at an angle of `45^@` with the horizontal. At the moment the ball is thrown, the player is 50 m from the coach. At what speed and in what direction must the player run to catch the ball at the same height at which it was released? `(g = 10 m//s^2)`.

Explanation

To solve the problem step by step, we will follow the projectile motion principles and use the given data.

### Step 1: Determine the Time of Flight
The time of flight \( T \) for a projectile is given by the formula:
\[ T = \frac{2u \sin \theta}{g} \]
where:
- \( u = 20 \, \text{m/s} \) (initial speed)
- \( \theta = 45^\circ \)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating \( T \):
\[ T = \frac{2 \times 20 \times \sin(45^\circ)}{10} \]
Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):
\[ T = \frac{40 \times \frac{1}{\sqrt{2}}}{10} = \frac{40}{10\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \, \text{s} \]

### Step 2: Calculate the Range of the Projectile
The range \( R \) of a projectile is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Calculating \( R \):
\[ R = \frac{20^2 \sin(90^\circ)}{10} \]
Since \( \sin(90^\circ) = 1 \):
\[ R = \frac{400}{10} = 40 \, \text{m} \]

### Step 3: Determine the Distance the Player Needs to Cover
The player is initially 50 m away from the coach. Since the ball will travel 40 m horizontally, the player must cover the remaining distance:
\[ \text{Distance to cover} = 50 \, \text{m} - 40 \, \text{m} = 10 \, \text{m} \]

### Step 4: Calculate the Speed of the Player
The speed \( v \) of the player can be calculated using the formula:
\[ v = \frac{\text{distance}}{\text{time}} \]
Substituting the values:
\[ v = \frac{10 \, \text{m}}{2\sqrt{2} \, \text{s}} \]
Calculating \( v \):
\[ v = \frac{10}{2\sqrt{2}} = \frac{5}{\sqrt{2}} \approx 3.535 \, \text{m/s} \]

### Step 5: Determine the Direction of the Player's Motion
The player must run towards the coach to catch the ball, which means the direction is towards the coach.

### Final Answer
The player must run at a speed of approximately \( 3.535 \, \text{m/s} \) towards the coach.

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