Question

A place-kicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.2 m/s at an angle of 51.0° to the horizontal.

(a) By how much does the ball clear or fall short (vertically) of clearing the crossbar? (Enter a negative answer if it falls short.)

Find the two initial velocity components, and from them and the equations of motion, determine the time and height when the football has traveled the required distance measured along the ground. m

(b) Does the ball approach the crossbar (and cross above or beneath it) while still rising or while falling?

rising

or falling    

Answer 1

Gravitational acceleration = g = -9.81 m/s²

Initial velocity of the ball = V₀ = 20.2 m/s

Angle of the initial velocity with the horizontal = = 51^o

Initial horizontal velocity = V_x0

V_x0 = VCos

V_x0 = 20.2Cos(51)

V_x0 = 12.71 m/s

Initial vertical velocity = V_y0

V_y0 = VSin

V_y0 = 20.2Sin(51)

V_y0 = 15.7 m/s

Horizontal distance of the crossbar from the ball = R = 36 m

Height of the crossbar = H = 3.05 m

Time taken by the ball to reach the crossbar = T

There is no force acting on the ball in the horizontal direction hence the velocity in the horizontal direction is constant.

R = V_x0T

36 = (12.71)T

T = 2.832 sec

Height of the ball when it reaches the crossbar = h

h = V_y0T + gT²/2

h = (15.7)(2.832) + (-9.81)(2.832)²/2

h = 5.12 m

Height the ball clears the crossbar by = H

H = h - H

H = 5.12 - 3.05

H = 2.07 m

Vertical velocity of the ball when it crosses the crossbar = V_y1

V_y1 = V_y0 + gT

V_y1 = 15.7 + (-9.81)(2.832)

V_y1 = -12.08 m/s

The negative velocity indicates the ball is traveling downwards.

The vertical velocity of the ball is negative when it crosses the crossbar hence the ball is falling when the ball crosses the crossbar.

a) The ball clears the crossbar by a distance of 2.07 m

b) The ball approaches the crossbar from above it and is falling when it crosses.