A group of students conducts an experiment in which they use
motion sensors to measure the velocity of a cart of mass M as it
rolls down a track inclined at an angle θ relative to horizontal.
The cart then makes a smooth transition onto a horizontal section
of the track and moves to the right along the track. At the end of
the horizontal section is an elastic bumper with which the cart
collides and rebounds. The graph of velocity as a function of time
for the entire motion of the cart is shown. There is negligible
friction between the cart and the track.
(a)
i. Derive an expression for the acceleration of the cart on the
inclined portion of the ramp. Express your answer in terms of θ, M,
and physical constants, as appropriate.
ii. Use your expression from part (a)(i) and data from the graph
to determine a numerical value for angle of the incline θ .
(b) The cart reaches the bumper with speed v1 and rebounds with
speed v2 . The cart is in contact with the bumper with a time Δt
.
i. Derive an expression for the magnitude of the average net
force F exerted on the cart by the bumper. Express your answer in
terms of M , v1 , v2 , Δt , and physical constants, as
appropriate.
ii. Is the force the bumper exerts on the cart directed to
the left or to the right?
iii. Assuming the mass of the cart is 0.25kg , use your
expression from (b)(i) and data from the graph to calculate the
value for the magnitude of the average net force on the cart during
the collision with the bumper.
(c) A student claims that friction is negligible in this
experiment. Do you agree or disagree with the student’s claim?
Agree Disagree Not enough information to agree or
disagree
Justify your answer.
: (d) Calculate the distance that the cart travels down the
inclined ramp.
: (e) On the axis below, sketch a graph of the acceleration a of
the cart as a function of time t. The time t1 is when the cart
transitions from the incline to the horizontal surface.
1.25
1.00
0.75 0.50 Velocity (m/s) 0.25 0.00 0.25
-0.50
0.75 0.250.500.751.001.251.501.752.002.252.50 0.00 Time (s)