Sports car problem

The sports car travels along a straight road such that its acceleration is described by the graph.

Specify the velocity of the car when s = 10 m. Express your answer to three significant figures and include the appropriate units.

From 0 to 10,

a = s

We know that

a ds = v dv

since a = s then

s ds = v dv

we integrate both sides

s²/2 = v²/2

which means that s = v, this only applies until 10 because the graph changes there.

so when s = 10 m then v = 10 m/s

Specify the velocity of the car when s = 15 m. Express your answer to three significant figures and include the appropriate units.

Now the acceleration changes, but notice that is constant. This can be solved with the formulas of dynamics for constant accelerations:

since the velocity at s = 10 is 10m/s,

V² = V₀² + 2*a*s

so V₀ = 10; a is constant at 10 m/s²; and s is final - initial distance which is 5 m

Plug in the numbers to solve for the final velocity to get:

V = 14.14 m/s

The right graph is easy to determine, remember that the v for the first 10 m is s with a slope of 1. Then remember that the acceleration is constant and positive for the rest: