Q4.7-44PE - College Physics (Urone)

Question

Integrated Concepts When starting a foot race, a 70.0-kg sprinter exerts an average force of 650 N backward on the ground for 0.800 s.

(a) What is his final speed?

(b) How far does he travel?

Step-by-Step Solution

Answer

(a) The final speed of the sprinter is 7.43 m/s.

(b) The distance traveled by the sprinter is 2.97 m.

Step 1:Given Data

• Mass of the sprinter = 70 kg.
• Time the force exerted = 0.800 s.
• Force exerted = 650 N.

Step 2: Determine the acceleration of the sprinter

Apply Newton’s Second Law of motion as:

 

$\mathit{F}\mathbf{=}\mathit{m}\mathit{a}$

 

Here, m is the mass of the sprinter, a is the acceleration, and F is the force exerted.

 

Substitute 70 kg for m and 650 N for F in the above expression, and we get,

 

$\begin{array}{c}\text{650\hspace{0.33em}N}=\text{70\hspace{0.33em}kg}\times a\\ a=\frac{650\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2}{70\text{\hspace{0.33em}kg}}\\ a=9.286{\text{\hspace{0.33em}m/s}}^2\end{array}$

Step 3: (a) Determine the final speed of the sprinter.

Apply the equation of motion as:

 

$\mathit{v}\mathbf{=}\mathit{u}\mathbf{+}\mathit{a}\mathit{t}$

 

Here, v is the final speed, u is the initial speed, and t is the time taken.

 

Substitute 0 m/s for u, 9.286 m/s² for a, and 0.8 s for t in the above equation, and we get,

 

$\begin{array}{c}v=0+9.286{\text{\hspace{0.33em}m/s}}^2\times 0.8\text{\hspace{0.33em}s}\\ =7.43\text{\hspace{0.33em}m/s}\end{array}$

 

Hence, the final speed of the sprinter is 7.43 m/s.

Step 4: (b) Determine the distance traveled by the sprinter

Apply the equation of motion as:

 

$\mathit{s}\mathbf{=}\mathit{u}\mathit{t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{a}{\mathit{t}}^{\mathbf{2}}$

 

Here, s is the distance traveled.

 

Substitute 0 m/s for u, 9.286 m/s² for a, and 0.8 s for t.

$\begin{array}{c}s=0+\frac{1}{2}\times 9.286{\text{\hspace{0.33em}m/s}}^{\text{2}}\times {0.8}^2{\text{\hspace{0.33em}s}}^{\text{2}}\\ =\frac{1}{2}\times 9.286{\text{\hspace{0.33em}m/s}}^{\text{2}}\times 0.64{\text{\hspace{0.33em}s}}^{\text{2}}\\ =\frac{1}{2}\times 5.94\text{\hspace{0.33em}m}\\ =2.97\text{\hspace{0.33em}m}\end{array}$ 

Hence, the distance traveled by the sprinter is 2.97 m.