In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints to a speed of 6.0 m/s in 1.5 s. Assuming that the player accelerates uniformly, determine the distance he runs.

This question aims to find the distance a basketball player runs from rest and moves with speed 6.0 m/s. The article uses an equation of motion to solve for unknown values. Equations of motion are mathematical formulas that describe a body’s position, velocity, or acceleration relative to a given frame of reference.

If the position of an object changes to a reference point, it is said to be in motion to that reference, whereas if it doesn’t change, it is at rest at that reference point. To better understand or solve different situations of rest and motion, we derive some standard equations related to the concepts of a body’s distance, displacement, velocity, and acceleration using an equation called the equation of motion.

Equations of motion

In the situation of motion with uniform or constant acceleration (with the same change in velocity in the same time interval), we derive the three standard equations of motion, also known as the laws of constant acceleration. These equations contain the quantities displacement(s), velocity (initial and final), time(t), and acceleration(s) that govern the motion of the particle. These equations can only be used when the acceleration of the body is constant, and the motion is a straight line. The three equations are:

The first equation of motion:

$$v=u+at$$

Second equation of motion:

$$F=ma$$

Third equation of motion:

$$v^2=u^2+2aS$$

Where:

1. $m$ is the mass
2. $F$ is the force 
3. $s$ is the total displacement
4. $u$ is the initial velocity
5. $v$ is the final velocity
6. $a$ is the acceleration 
7. $t$ represents the time of motion

Expert Answer

Since the sprinter accelerates uniformly, we can use the equation of motion. First, we need to calculate the sprinter’s acceleration using the first equation of motion:

$$v=u+at$$

$v$ is final velocity, and $u$ represents the initial velocity.

$$a={\displaystyle \frac{v-u}{t}}$$

$$a={\displaystyle \frac{6-0}{1.5}}$$

$$a=4{\displaystyle \frac{m}{s^2}}$$

Now the distance covered by the sprinter is calculated according to the $3rd$ equation of motion.

$$v^2=u^2+2aS$$

Rearrange the equation for the unknown $S$.

$$S={\displaystyle \frac{v^2-u^2}{2a}}$$

Plug values into the above equation to find the distance.

$$S={\displaystyle \frac{6^2-0}{2\times 4}}$$

$$S=4.5m$$

Hence, the distance run by the sprinter is $S=4.5m$.

Numerical Result

The distance run by the sprinter is $S=4.5m$.

Example

As a basketball player prepares to shoot the ball, he starts from rest and sprints at $8.0{\displaystyle \frac{m}{s}}$ in $2s$. Assuming the player accelerates uniformly, determine the distance he runs.

Solution

Since the sprinter accelerates uniformly, we can use the equation of motion. First, we need to calculate the sprinter’s acceleration using the first equation of motion:

$$v=u+at$$

$v$ is final velocity, and $u$ is the initial velocity.

$$a={\displaystyle \frac{v-u}{t}}$$

$$a={\displaystyle \frac{8-0}{2}}$$

$$a=4{\displaystyle \frac{m}{s^2}}$$

Now the distance covered by the sprinter is calculated according to the $3rd$ equation of motion:

$$v^2=u^2+2aS$$

Rearrange the equation for the unknown $S$.

$$S={\displaystyle \frac{v^2-u^2}{2a}}$$

Plug values into the above equation to find the distance.

$$S={\displaystyle \frac{8^2-0}{2\times 4}}$$

$$S=8m$$

Hence, the distance run by the sprinter is $S=8m$.

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