Question

a football is kickes off with a intial speed of 19.6m to have the maximum range.goal keeper standing on the goal line 67.4m away in the direction of the kick starts running opposite to the direction to the direction of kick to meet the ball at that instant.what must be the speed his speed is.if he is to catch the ball before hits the ground​

Answer 1

Answer: Velocity of man = 10 $\frac{m}{s}$

Explanation:

Football kicked will follow a projectile motion.

Maximum Range of projectile motion = R = $\frac{u^2Sin2(theta)}{g}$

It is given in question that ball kicked for maximum range.

for maximum range, sin2θ shall be equal to 1.

R = $\frac{u^2}{g}$   = $\frac{19.6^2}{9.8}$ = 39.2 Metres is the maximum range.

∴ Total distance the man should run = 67.4 - 39.2 = 28.2 metre

Let v be the velocity of the man.

∴Time t taken by man will be equal to the time taken by the ball to reach before it hits the ground.

Time taken for projectile motion = $\frac{2u.Sin(theta)}{g}$

= $\frac{2 X 19.6 X sin45}{9.8}$ = $\frac{4}{\sqrt{2} }$ = 2$\sqrt{2}$ = 2.82 s

Velocity of man = $\frac{Distance}{Speed}$ = $\frac{28.2}{2.82}$ = 10 $\frac{m}{s}$

∴Velocity of man = 10 $\frac{m}{s}$

Answer 2

The speed his speed is if he is to catch the ball before hits the ground​ is 10 m/s.

Explanation:

The football undergoes projectile motion.

The maximum of the ball is given by the formula:

R = (u² sin 2θ)/g

Where,

u = Initial speed = 19.6 m/s (given)

g = Acceleration due to gravity = 9.8 m/s² (constant)

On substituting the values, we get,

R = ((19.6)² sin 90)/9.8

∴ R = 39.2 m

Now, the distance the traveled by the kicker to catch the ball is:

D = 67.4 m - 39.2 m = 28.2 m

The time taken by the ball to reach the ground is given by the formula:

t = (2u sin θ)/g

On substituting the values, we get,

t = (2 × 19.6 × sin 45°)/9.8

t = 4/√2

∴ t = 2.82 s = Time taken by keeper to reach the ball

Now, the velocity is given by the formula:

Velocity = Distance/Time = D/t

On substituting the values, we get,

V = 28.2/2.82

∴ V = 10 m/s