law of conservation

Question

A man on a motorcycle plans to make a jump a ramp.

If he leaves the ramp with a speed of 35.0 m/s and has a speed of 33.5 m/s at the top of his trajectory, determine his maximum height (h) (in m) above the end of the ramp. Ignore friction and air resistance.

Answer 1

Let's start by listing everything that we do and don't know.

v₁ = 35 m/s

h₁ = 0 m

v₂ = 33.5 m/s

h₂ = ?

g = 9.81 m/s²

Now lets look at the man's energy at both position 1 (at the top of the ramp) and 2 (at the top of his trajectory). From the conservation of energy law, we know that the total energy he has at both points must be equal. His total energy is made up of both kinetic and potential energy.

Recall that the equation for kinetic energy is KE = 1/2*m*v², and the equation for potential energy is PE = m*g*h where m is mass, v is velocity, g is the gravitational acceleration (9.81 m/s² on earth), and h is the height.

Setting his total energy at both points equal leads to

E₁ = E₂

KE₁ + PE₁ = KE₂ + PE₂

1/2*m*v₁² + m*g*h₁ = 1/2*m*v₂² + m*g*h₂

We know all of the terms except for h₂, so lets rearrange to solve for h₂.

h₂ = (1/2*m*v₁² + m*g*h₁ - 1/2*m*v₂²) / m*g

Using some algebra and simplifying leaves

h₂ = 1/2*v₁²/g + h₁ - 1/2*v₂²/g

Now plug in the values that we know

h₂ = 1/2*(35 m/s)²/ (9.81 m/s²) + 0 - 1/2*(33.5² m/s) / (9.81 m/s²)

h₂ = 5.23 m

The last question to ask is: does this answer make sense and is it reasonable? I personally don't have a good feel for how tall 5.23 m is, or how fast 35 m/s is, so let's convert to units that I have more of an intuitive feel for. Converting to imperial units gives us 17.1 feet and 78.2 mph. Does it seem reasonable that a daredevil going over a ramp with a motorcycle at nearly 80 mph can jump 17 feet in the air? I'd say that doesn't sound too unreasonable, especially given that we're ignoring friction and air resistance. But I wouldn't want to be the one to try out this experiment at home!