A player kicks a football at an angle of 45 degree with initial speed of 20 metre per second a second player on the goal line 60 metre away in the direction of a kick start training to receive the ball at that instant find the constant speed of the second player with which he should run to catch the ball before he hit the ground
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34.09m/s
To find the constant speed at which the second player should run to catch the ball before it hits the ground, we need to consider the motion of the ball and the relative motion of the second player.
Here are the steps to determine the constant speed:
Step 1: Break the initial velocity of the ball into its horizontal and vertical components.
The initial speed of the ball is given as 20 meters per second at an angle of 45 degrees. The vertical component of the velocity will be 20 * sin(45) and the horizontal component will be 20 * cos(45).
Vertical component: V_y = 20 * sin(45) = 20 * √2 / 2 = 10√2
Horizontal component: V_x = 20 * cos(45) = 20 * √2 / 2 = 10√2
Step 2: Analyze the vertical motion of the ball.
The vertical motion of the ball can be described using the equations of motion under constant acceleration. In this case, the only force affecting the ball is gravity, which will cause the ball to accelerate downwards with an acceleration of -9.8 m/s^2.
Using the vertical motion equation:
y = V_y * t + (1/2) * a * t^2
Here, y represents the vertical distance, V_y is the initial vertical velocity, t is the time, and a is the acceleration (gravity).
Since the ball will hit the ground when y = 0, we can solve for the time it takes for the ball to hit the ground.
0 = 10√2 * t - (1/2) * 9.8 * t^2
Simplifying this equation, we get:
0 = 5√2 * t - 4.9 * t^2
Step 3: Solve for the time it takes for the ball to hit the ground.
This is a quadratic equation, so let's solve it by factoring or using the quadratic formula.
0 = 5√2 * t - 4.9 * t^2
0 = t(5√2 - 4.9t)
Setting each factor equal to zero:
t = 0 or 5√2 - 4.9t = 0
The first solution (t = 0) is not meaningful in this context, so we can disregard it.
Solving for t in the second equation:
5√2 - 4.9t = 0
5√2 = 4.9t
t = (5√2) / 4.9
Step 4: Calculate the horizontal distance traveled by the ball in this time.
The horizontal motion of the ball is constant, so we can use the equation:
x = V_x * t
Substituting the known values:
x = (10√2) * (5√2) / 4.9
Step 5: Determine the constant speed of the second player.
In order to catch the ball before it hits the ground, the second player needs to reach the point where the ball will land. This point is given as 60 meters away from the kick start.
Therefore, the constant speed at which the second player should run can be calculated using the formula:
Speed = Distance / Time
Speed = 60 meters / [ (5√2) / 4.9 ]
Simplifying this expression:
Speed = 60 * 4.9 / (5√2)
Calculating the value using a calculator, we find:
Speed ≈ 12.18 meters per second
So, the constant speed at which the second player should run to catch the ball before it hits the ground is approximately 12.18 meters per second.