A soccer ball is kicked from the ground with an initialspeed of 19.5 m/s at an upward angle of 45°. A player 55 m awayin the direction of the kick starts running to meet the ball at thatinstant. What must be his average speed if he is to meet the balljust before it hits the ground?
Question
A soccer ball is kicked from the ground with an initialspeed of 19.5 m/s at an upward angle of 45°. A player 55 m awayin the direction of the kick starts running to meet the ball at thatinstant. What must be his average speed if he is to meet the balljust before it hits the ground?
Solution
To solve this problem, we need to find out two things:
- How long the ball is in the air (time of flight).
- The speed at which the player needs to run to reach the ball in that time.
Step 1: Calculate the time of flight
The time of flight of a projectile launched with an initial speed (u) at an angle (θ) is given by the formula:
t = 2u*sin(θ) / g
where:
- u is the initial speed of the ball (19.5 m/s),
- θ is the angle of projection (45°), and
- g is the acceleration due to gravity (9.8 m/s²).
First, we need to convert the angle from degrees to radians because the trigonometric functions in most calculators use radians. 45 degrees is equal to 45*(π/180) = 0.7854 radians.
Then, we substitute the values into the formula:
t = 219.5sin(0.7854) / 9.8 t = 2.828 seconds
Step 2: Calculate the player's speed
The player's speed (v) is given by the distance he needs to cover (d) divided by the time it takes (t):
v = d / t
where:
- d is the distance the player needs to cover (55 m), and
- t is the time of flight of the ball (2.828 seconds).
Substituting the values into the formula, we get:
v = 55 / 2.828 v = 19.45 m/s
So, the player must run at an average speed of 19.45 m/s to meet the ball just before it hits the ground.
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