Question

In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a skier who skies 70.0 m along a 30° slope neglecting friction:

(a) Starting from rest.

(b) Starting with an initial speed of 2.50 m/s.

(c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events.

Short answer

(a) The final speed of the skier is 26.19 m/s and the time taken to reach bottom is 5.34 s.

(b) The final speed of the skier is 26.31 m/s and the time taken to reach bottom is 4.86 s.

(c) Yes, it is advantageous to get a running start because it takes short time to finish the competitive events.

Step-by-step solution

Conservation of energy

Schematic diagram is shown below:

Skier skies downhills

Here,$v_i$is the initial velocity of the skier, d is the length of the track (d = 70.0 m), and h is the height of the hill.

The height of the hill is,

$h=d\sin \left(30°\right)$

Putting all known values,

$$\begin{gathered} h=\left(70.0\text{m}\right)\times \sin \left(30°\right) \\ =35\text{m} \end{gathered}$$

Conservation of energy: The universe's total energy is conserved. According to conservation of energy, energy neither be created not be destroyed, only the form of energy can be changed.

From the conservation of energy,

$$\begin{gathered} {\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}={\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}} \\ \frac{1}{2}m{v}_f^2+mg{h}_f=\frac{1}{2}m{v}_i^2+mg{h}_i \end{gathered}$$ (1.1)

Here, m is the mass of the skier,$v_f$is the final velocity of the skier, g is the acceleration due to gravity$\left(g=9.8\text{m}/{\text{s}}^2\right)$,is the final height ($h_f=0$as the siker is at the ground),$v_i$is the initial velocity of the skier, and$h_i$is the initial height of skier ($h_i=h=35\text{m}$as the skier starts from the top of the hill).

Rearranging equation (1) in order to get an expression for final velocity of the skier,

$v_f=\sqrt{v_i^2+2g\left(h_i-h_f\right)}$ (1.2)

Third equation of motion is given as,

$v_f^2=v_i^2+2ad$ (1.3)

Here, a is the acceleration of skier.

Rearranging equation (1.3) in order to get an expression for acceleration,

$a=\frac{v_f^2-v_i^2}{2d}$ (1.4)

First equation of motion is given as,

$v_f=v_i+at$ (1.5)

Here, t is the time taken to reach the ground.

Rearranging equation (1.5) in order to get an expression for time,

$t=\frac{v_f-v_i}{a}$ (1.6)

The final speed and the time taken for a skier who starts from the rest

$$\begin{gathered} t=\frac{\left(26.19\text{m}/\text{s}\right)-\left(0\right)}{\left(4.9\text{m}/{\text{s}}^2\right)} \\ =5.34\text{s} \end{gathered}$$(a)

When the skier starts from rest i.e. $v_i=0$. The final velocity can be calculated using equation (1.2).

Putting all known values in equation (1.2),

$$\begin{gathered} v_f=\sqrt{{\left(0\right)}^2+2\times \left(9.8\text{m}/{\text{s}}^2\right)\times \left[\left(35.0\text{m}\right)-\left(0\right)\right]} \\ =26.19\text{m}/\text{s} \end{gathered}$$

As a result, the final velocity of the skier when he starts from the rest is 26.19 m/s.

The acceleration of the skier can be calculated using equation (1.4).

Putting all known values in equation (1.4),

$$\begin{gathered} a=\frac{{\left(26.19\text{m}/\text{s}\right)}^2-{\left(0\right)}^2}{2\times \left(70\text{m}\right)} \\ =4.9\text{m}/{\text{s}}^2 \end{gathered}$$

Time taken to reach the ground can be calculated using equation (1.6).

Putting all known values in equation (1.6),

Therefore, the required time taken by the skier when he starts from the rest is 5.34 s.

The final speed and the time taken by a skier starts with some initial velocity

(b)

When the skier starts with some initial velocity$v_i=2.50\text{m}/\text{s}$. The final velocity can be calculated using equation (1.2).

Putting all known values in equation (1.2),

$$\begin{gathered} v_f=\sqrt{{\left(2.50\right)}^2+2\times \left(9.8\text{m}/{\text{s}}^2\right)\times \left[\left(35.0\text{m}\right)-\left(0\right)\right]} \\ =26.31\text{m}/\text{s} \end{gathered}$$

Hence, the final velocity of the skier when he starts with some initial velocity is 26.31 m/s.

The acceleration of the skier can be calculated using equation (1.4).

Substituting all known values in equation (1.4),

$$\begin{gathered} a=\frac{{\left(26.31\text{m}/\text{s}\right)}^2-{\left(2.50\right)}^2}{2\times \left(70\text{m}\right)} \\ =4.9\text{m}/{\text{s}}^2 \end{gathered}$$

Time taken to reach the ground can be calculated using equation (1.6).

Putting all known values in equation (1.6),

$$\begin{gathered} t=\frac{\left(26.31\text{m}/\text{s}\right)-\left(2.50\right)}{\left(4.9\text{m}/{\text{s}}^2\right)} \\ =4.86\text{s} \end{gathered}$$

Therefore, the required time taken by the skier when he starts from the rest is 4.86 s.

Advantages of running start in competitive events.

(c)

In competitive events small fraction of time plays an important role, and as we can observe from the answer to part (b), a running start takes shorter time. Thus, it is very advantageous to get a running start in competitive events.