3. (20 pts) A sprinter in a 100 m dash accelerates for the first 5 s of the race

Question

3. (20 pts) A sprinter in a 100 m dash accelerates for the first 5 s of the race, when he reaches his top speed of 10 m/s. He maintains his speed for 3 s, then decelerates to 9 m/s at the finish line. What were the average accelerations during the three phases of the race? What was his average velocity for the whole race? 4. (20 pts) A soccer player punts a ball 25 m down the field. If the kick initially has an angle of 30° relative to vertical and height of 1 m when it leaves the kicker's foot, what is the initial velocity required for the kick to travel the measured distance?

Explanation / Answer

3)

Let the distance traveled during the acceleration phase of the race be d1.

Now, using the equation of motion:

v = u + at

So,10 = 0 + a*5

So, a = 2 m/s2

So, distance traveled , d1 = 0.5*at^2 = 0.5*2*5^2 = 25 m

Distance traveled while at constant speed, d2 = 10*3 = 30 m

So, distance traveled while decelerating, d3 = 100 - 25 - 30 = 45 m

So, using the equation of motion:

v^2 = u^2 + 2as

So, 9^2 = 10^2 + 2*a*45

So, a = -0.21 m/s2

So, time taken in this phase = (v- u)/a = (9 - 10)/(-0.21) = 4.76 s

So, average acceleration during 1st phase = 2 m/s2 <------ solved above

average acceleration during 2nd phase = 0 <---- as the velocity is constant

Average acceleration during 3rd phase = -0.21 m/s2

Average velocity for the whole race = distance traveled / total time taken

= 100/(5 + 3 + 4.76)

= 7.84 m/s

4)

Using the equation of motion :

s = ut + 0.5*at^2

s = displacement = 25 i - j <------ 25 m horizontally and 1m vertically down

a = -9.8 j m/s2

u = x i + tan(30 deg)*x j <---- Here x = =horizontal velocity of projection ... as the angle of projection is equal to 30 deg

25 i - j = (x i + tan(30 deg)*x j )*t + 0.5*(-9.8 j)*t^2

So, 25 = x*t and -1 = tan(30 deg)*x*t - 4.9*t^2

Solving the two equations, we get :

x = 14.1 m/s

t = 1.78 s

So, initial velocity , u = 14.1 i + 14.1*tan(30 deg) j

= 14.1 i + 8.1 j

So, | u | = sqrt(14.1^2 + 8.1^2)

= 16.3 m/s <-------- answer

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