Answer:
a. 250 kg m/s upwards
b. 5000 N
c. The force on the gymnast is five times higher when she lands on the driveway.
Explanation:
The speed of a falling object can be found using constant acceleration kinematics or conservation of energy. When the gymnast lands on the gym mat, the mat exerts a force on the gymnast that stops their fall. The impulse of this force is equal to the change in momentum, or the force times the amount of time.
a. First, we need to find the gymnast's speed when she lands on the mat. We can do this using kinematics or energy. Taking down to be positive, if we use kinematics:
v² = u² + 2as
v² = (0 m/s)² + 2 (9.8 m/s²) (2.0 m)
v = 6.26 m/s
Alternatively, if we use energy:
PE = KE
mgh = ½ mv²
v² = 2gh
v² = 2 (9.8 m/s²) (2.0 m)
v = 6.26 m/s
Next, the impulse is equal to the change in momentum.
J = Δp
J = mΔv
J = (40 kg) (0 m/s − 6.26 m/s)
J = -250 kg m/s
The impulse is 250 kg m/s upwards.
b. The impulse is equal to the force times the time.
J = FΔt
250 kg m/s = F (0.05 s)
F = 5000 N
c. This time, the gymnast comes to a stop in 0.01 seconds.
J = FΔt
250 kg m/s = F (0.01 s)
F = 25,000 N
The force is five times higher when the gymnast lands on the driveway. This higher force causes the gymnast to sprain her ankle.