Question

1. a wide receiver runs with the ball at a constant 9 yards per second (9 yd/s) past a cornerback that is standing still at the 10 yard line of a football field (the receiver has 90 yards to go to score a touchdown). if the cornerback accelerates at 4.9 yd/s² for two seconds to reach his top speed, and then runs at that constant top speed, will the cornerback catch the wide receiver before he scores? 6. if the cornerback in problem 5 does not catch the receiver, how close to the receiver is he when the receiver scores? if he does catch him, how far did he run in order to catch him?

Explanation:

Step1: Calculate cornerback's top - speed

Use the formula $v = v_0+at$. Since $v_0 = 0$ (starts from rest), $a = 4.9$ yd/s² and $t = 2$ s.
$v=0 + 4.9\times2=9.8$ yd/s

Step2: Calculate distance the cornerback runs during acceleration

Use the formula $d_1=v_0t+\frac{1}{2}at^{2}$. With $v_0 = 0$, $a = 4.9$ yd/s² and $t = 2$ s.
$d_1=0\times2+\frac{1}{2}\times4.9\times2^{2}=9.8$ yards

Step3: Calculate time for the wide - receiver to score

The wide - receiver has $d_w=90$ yards to go and a speed of $v_w = 9$ yd/s. Using $t_w=\frac{d_w}{v_w}$.
$t_w=\frac{90}{9}=10$ s

Step4: Calculate time the cornerback runs at constant speed

The cornerback accelerates for $t_1 = 2$ s. So the time he runs at constant speed is $t_2=t_w - 2=10 - 2 = 8$ s.

Step5: Calculate distance the cornerback runs at constant speed

Using $d_2=v\times t_2$, with $v = 9.8$ yd/s and $t_2 = 8$ s.
$d_2=9.8\times8 = 78.4$ yards

Step6: Calculate total distance the cornerback runs

$d_c=d_1 + d_2=9.8+78.4 = 88.2$ yards

Since $d_c=88.2<90$, the cornerback does not catch the wide - receiver before he scores.

Step7: Calculate how close the cornerback is when the wide - receiver scores

The distance between them is $\Delta d=90 - 88.2=1.8$ yards

Answer:

The cornerback does not catch the wide - receiver before he scores. The cornerback is 1.8 yards away from the wide - receiver when the wide - receiver scores.