Question

In: Physics

A sprinter accelerates at 3.08 m/s2 for a certain number of seconds during a 100.0 meter...

A sprinter accelerates at 3.08 m/s² for a certain number of seconds during a 100.0 meter race. She runs the rest of the race at top speed and finishes the race in 10.59 s. What is her top speed?

Expert Solution

Length of the race = L = 100 m

Total time taken by the sprinter to finish the race = T = 10.59 sec

Initial speed of the sprinter = V₁ = 0 m/s

Acceleration of the sprinter = a = 3.08 m/s²

Top speed of the sprinter = V₂

Time taken to reach the top speed = T₁

V₂ = V₁ + aT₁

V₂ = 0 + aT₁

V₂ = aT₁

Distance the sprinter travels while accelerating = L₁

L₁ = V₁T₁ + aT₁²/2

L₁ = (0)T₁ + aT₁²/2

L₁ = aT₁²/2

Time period the sprinter travels at top speed = T₂

T = T₁ + T₂

T₂ = T - T₁

Distance the sprinter travels at top speed = L₂

L₂ = V₂T₂

L₂ = (aT₁)(T - T₁)

L₂ = aTT₁ - aT₁²

L = L₁ + L₂

L = aT₁²/2 + aTT₁ - aT₁²

L = aTT₁ - aT₁²/2

100 = (3.08)(10.59)T₁ - (3.08)T₁²/2

100 = 32.62T₁ - 1.54T₁²

1.54T₁² - 32.62T₁ + 100 = 0

T₁ = 17.46 sec or 3.72 sec

T₁ cannot be greater than the total time period which is 10.59 sec.

T₁ = 3.72 sec

V₂ = aT₁

V₂ = (3.08)(3.72)

V₂ = 11.46 m/s

Top speed of the sprinter = 11.46 m/s

genius_generous answered 5 months ago

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